#7148260

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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
prime radical	#45196	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1787 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)(13,14,18,23,19,24,15,20)(16,27,25,36,29,35,32,21)(17,22,30,37,31,38,26,33)(28,40,39,43,42,44,41,34);(1,2)(3,4)(6,9)(7,11)(13,15,19,18)(14,20,24,23)(16,25,29,32)(17,26,31,30)(21,27,36,35)(22,33,38,37)(28,39,42,41)(34,40,43,44);(5,6)(7,10)(8,9)(11,12)(13,16,19,29)(14,21,24,36)(15,25,18,32)(17,28,31,42)(20,27,23,35)(22,34,38,43)(26,39,30,41)(33,40,37,44);(2,4)(6,9)(10,12)(13,17)(14,22)(15,26)(16,28)(18,30)(19,31)(20,33)(21,34)(23,37)(24,38)(25,39)(27,40)(29,42)(32,41)(35,44)(36,43);(5,7,8,11)(6,10,9,12)(13,15,19,18)(14,23,24,20)(16,25,29,32)(17,30,31,26)(21,35,36,27)(22,33,38,37)(28,41,42,39)(34,40,43,44)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7055456);