#7006915

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2057 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)(13,14,20,35,19,24,18,23)(15,25,17,21,27,41,28,38)(16,29,36,40,32,22,33,44)(26,43,34,39,30,42,31,37);(5,6)(7,11)(8,10)(9,12)(13,15,19,27)(14,21,24,38)(16,30,32,26)(17,20,28,18)(22,39,29,37)(23,41,35,25)(31,36,34,33)(40,42,44,43);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,32)(14,22,24,29)(15,26,27,30)(17,34,28,31)(18,33,20,36)(21,37,38,39)(23,40,35,44)(25,42,41,43);(7,12)(9,11)(13,17,18,15,19,28,20,27)(14,21,23,25,24,38,35,41)(16,31,33,30,32,34,36,26)(22,39,40,43,29,37,44,42);(6,10)(9,11)(15,27)(16,32)(17,28)(21,38)(22,29)(25,41)(33,36)(40,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=f,f^2=i,b^a=bf,c^a=cg,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, 6914111);