#7145944

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

2189 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)(7,11)(10,12)(13,14)(15,19)(16,27)(17,20)(18,22)(21,28)(23,30)(24,33)(25,39)(26,34)(29,36)(31,38)(32,37)(35,41)(40,42)(43,44);(2,3)(6,9)(7,11)(13,15,18,24)(14,19,22,33)(16,25,28,37)(17,26,30,38)(20,34,23,31)(21,32,27,39)(29,40,41,44)(35,43,36,42);(5,6)(7,10)(8,9)(11,12)(14,20)(15,24)(16,28)(17,30)(19,31)(21,36)(22,23)(27,35)(32,43)(33,34)(39,42)(40,44);(1,3,4,2)(6,9)(10,12)(13,16)(14,21)(15,25)(17,29)(18,28)(19,32)(20,35)(22,27)(23,36)(24,37)(26,40)(30,41)(31,42)(33,39)(34,43)(38,44);(5,7,8,11)(6,10,9,12)(14,22)(15,24)(16,28)(20,23)(26,38)(29,41)(32,39)(42,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,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, 7053140);