#7062892

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#24435	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1885 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6970088);