#7148197

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1760 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)(16,27,25,35,29,38,32,21)(17,22,31,37,30,36,26,33)(28,40,39,43,41,44,42,34);(1,2)(3,4)(6,9)(7,11)(13,15,19,18)(14,20,24,23)(16,25,29,32)(17,26,30,31)(21,27,35,38)(22,33,36,37)(28,39,41,42)(34,40,43,44);(5,6)(7,10)(8,9)(11,12)(13,16,19,29)(14,21,24,35)(15,25,18,32)(17,28,30,41)(20,27,23,38)(22,34,36,43)(26,39,31,42)(33,40,37,44);(2,4)(6,9)(10,12)(13,17,19,30)(14,22,24,36)(15,26,18,31)(16,28,29,41)(20,33,23,37)(21,34,35,43)(25,39,32,42)(27,40,38,44);(5,7,8,11)(6,10,9,12)(17,30)(22,36)(26,31)(28,41)(33,37)(34,43)(39,42)(40,44)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7055393);