#484213

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	#23438	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

1069 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1069 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)(11,23)(12,28)(13,32)(14,22)(15,21)(16,19)(17,25)(18,30)(20,35)(24,51)(26,57)(27,50)(29,60)(31,47)(33,48)(34,44)(36,46)(37,42)(38,45)(39,66)(40,56)(41,64)(43,63)(49,53)(52,58)(54,71)(55,69)(59,62)(61,67)(65,70)(68,72);(1,2)(3,5)(4,6)(7,8)(9,11,16,27)(10,17,19,40)(12,24,31,55)(13,33,36,64)(14,26,15,38)(18,39,42,67)(20,45,44,57)(21,41,22,48)(23,46,50,32)(25,34,56,35)(28,58,47,68)(29,61,63,66)(30,53,37,54)(43,69,60,51)(49,70,71,59)(52,65,72,62);(1,3)(2,5)(4,7)(6,8)(9,12)(10,18)(11,24)(13,29)(14,37)(15,30)(16,31)(17,39)(19,42)(20,43)(21,47)(22,28)(23,49)(25,52)(26,54)(27,55)(32,59)(33,61)(34,65)(35,62)(36,63)(38,53)(40,67)(41,68)(44,60)(45,69)(46,70)(48,58)(50,71)(51,57)(56,72)(64,66);(10,19)(17,40)(18,42)(20,44)(21,22)(23,50)(28,47)(32,46)(39,67)(41,48)(43,60)(45,57)(49,71)(51,69)(58,68)(59,70)

Polycyclic presentation:

2^9|b^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, 391409);