#484216

order	512
nilpotency class	4
isoclinism	isoclinic class of #484212
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60749	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1075 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1075 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(5,8)(9,10,16,22)(11,23,27,51)(12,28,32,43)(13,21,15,19)(14,34,37,46)(17,39,42,26)(18,29,45,31)(20,33,48,36)(24,52,56,70)(25,57,38,47)(30,61,64,44)(35,49,65,41)(40,62,68,72)(50,53,71,55)(54,58,66,67)(59,60,69,63);(1,2)(3,5)(4,6)(7,8)(9,11)(10,17)(12,24)(13,25)(14,35)(15,38)(16,27)(18,40)(19,41)(20,47)(21,49)(22,42)(23,34)(26,36)(28,58)(29,53)(30,62)(31,55)(32,56)(33,39)(37,65)(43,67)(44,70)(45,68)(46,51)(48,57)(50,59)(52,61)(54,63)(60,66)(64,72)(69,71);(1,3)(2,5)(4,7)(6,8)(9,12)(10,18)(11,24)(13,29)(14,30)(15,31)(16,32)(17,40)(19,43)(20,44)(21,28)(22,45)(23,50)(25,53)(26,54)(27,56)(33,60)(34,59)(35,62)(36,63)(37,64)(38,55)(39,66)(41,67)(42,68)(46,69)(47,70)(48,61)(49,58)(51,71)(52,57)(65,72);(9,13,16,15)(10,19,22,21)(11,25,27,38)(12,29,32,31)(14,33,37,36)(17,41,42,49)(18,43,45,28)(20,46,48,34)(23,47,51,57)(24,53,56,55)(26,35,39,65)(30,60,64,63)(40,67,68,58)(44,69,61,59)(50,70,71,52)(54,62,66,72)

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391412);