#7006891

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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

2095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fgi,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

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, 6914087);