#505700

order	512
nilpotency class	4
isoclinism	isoclinic class of #500843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

489 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ehi,d^2=eh,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412896);