#7006414

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2032 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6913610);