#7195625

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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

2450 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)(13,14,18,22,19,23,31,37)(15,24,26,33,27,36,17,20)(16,21,30,35,29,34,41,43)(25,38,40,42,39,44,28,32);(6,9)(7,11)(13,15,19,27)(14,20,23,33)(16,25,29,39)(17,31,26,18)(21,32,34,42)(22,36,37,24)(28,41,40,30)(35,44,43,38);(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,16)(14,21)(15,25)(17,28)(18,30)(19,29)(20,32)(22,35)(23,34)(24,38)(26,40)(27,39)(31,41)(33,42)(36,44)(37,43);(6,9)(10,12)(13,17,18,15,19,26,31,27)(14,20,22,24,23,33,37,36)(16,28,30,25,29,40,41,39)(21,32,35,38,34,42,43,44);(5,7,8,11)(6,10,9,12)(16,29)(21,34)(25,39)(28,40)(30,41)(32,42)(35,43)(38,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=f,e^2=h,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

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, 7102821);