#7195673

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2424 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7102869);