#7145773

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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

2163 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)(7,11)(10,12)(13,14,17,22,18,23,21,37)(15,24,27,33,28,26,31,19)(16,20,30,36,29,35,34,44)(25,38,41,43,40,39,42,32);(6,9)(7,11)(13,15,18,28)(14,19,23,33)(16,25,29,40)(17,31,21,27)(20,32,35,43)(22,26,37,24)(30,42,34,41)(36,39,44,38);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,25)(17,30)(18,29)(19,32)(21,34)(22,36)(23,35)(24,38)(26,39)(27,41)(28,40)(31,42)(33,43)(37,44);(6,9)(10,12)(13,14,21,37,18,23,17,22)(15,26,31,33,28,24,27,19)(16,20,34,44,29,35,30,36)(25,39,42,43,40,38,41,32);(5,7,8,11)(6,10,9,12)(16,29)(20,35)(25,40)(30,34)(32,43)(36,44)(38,39)(41,42)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,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, 7052969);