#7065177

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1792 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)(13,14,20,17,19,23,18,24)(15,25,33,21,28,38,27,36)(16,29,34,37,31,22,32,30)(26,40,44,43,41,39,42,35);(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,21,23,36)(16,26,31,41)(17,25,24,38)(18,33,20,27)(22,35,29,43)(30,39,37,40)(32,44,34,42);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,22)(15,26)(17,30)(18,32)(19,31)(20,34)(21,35)(23,29)(24,37)(25,39)(27,42)(28,41)(33,44)(36,43)(38,40);(7,12)(9,11)(13,17,20,23,19,24,18,14)(15,21,33,38,28,36,27,25)(16,30,34,29,31,37,32,22)(26,35,44,40,41,43,42,39);(6,10)(9,11)(14,23)(16,31)(17,24)(21,36)(25,38)(26,41)(32,34)(42,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6972373);