#7611720

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	#45412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1999 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)(7,9)(8,10)(11,12)(13,14,19,24)(15,22,28,40)(16,21,29,36)(17,32,27,20)(18,34,25,23)(26,39,42,37)(30,35,33,44)(31,43,41,38);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,25,17,19,28,18,27)(14,20,34,40,24,32,23,22)(16,26,41,33,29,42,31,30)(21,35,43,37,36,44,38,39);(6,10)(7,11)(13,16)(14,21)(15,26)(17,33)(18,31)(19,29)(20,35)(22,39)(23,38)(24,36)(25,41)(27,30)(28,42)(32,44)(34,43)(37,40);(1,2,3,4)(6,10)(9,12)(16,29)(21,36)(26,42)(30,33)(31,41)(35,44)(37,39)(38,43);(13,17,25,28,19,27,18,15)(14,22,34,20,24,40,23,32)(16,30,41,26,29,33,31,42)(21,37,43,44,36,39,38,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=h,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^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, 7518916);