#7611941

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45241	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2025 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,22)(15,19,24,31)(16,20,28,35)(17,21,29,36)(23,30,27,34)(25,32,38,42)(26,33,40,44)(37,41,39,43);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(16,26)(17,29)(20,33)(21,36)(23,37)(25,38)(27,39)(28,40)(30,41)(32,42)(34,43)(35,44);(6,10)(7,11)(13,15)(14,19)(16,27)(17,25)(18,24)(20,34)(21,32)(22,31)(23,28)(26,39)(29,38)(30,35)(33,43)(36,42)(37,40)(41,44);(6,10)(9,12)(13,16,17,26,18,28,29,40)(14,20,21,33,22,35,36,44)(15,23,25,37,24,27,38,39)(19,30,32,41,31,34,42,43);(1,3,4,2)(15,24)(19,31)(23,27)(25,38)(30,34)(32,42)(37,39)(41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei

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, 7519137);