#7527552

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#45236	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1998 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=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, 7434748);