#6617871

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1754 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)(13,14,19,24)(15,20,28,36)(16,29,31,37)(17,22,18,23)(21,32,38,30)(25,39,42,44)(26,34,27,35)(33,40,43,41);(1,2,3,4)(6,10)(8,11)(13,15)(14,20)(16,25)(17,27)(18,26)(19,28)(21,33)(22,35)(23,34)(24,36)(29,39)(30,41)(31,42)(32,40)(37,44)(38,43);(5,6)(7,8)(9,10)(11,12)(13,16,19,31)(14,21,24,38)(15,25,28,42)(17,30,18,32)(20,33,36,43)(22,37,23,29)(26,40,27,41)(34,39,35,44);(5,7,9,12)(6,11,10,8)(13,17,19,18)(14,22,24,23)(15,26,28,27)(16,30,31,32)(20,34,36,35)(21,37,38,29)(25,40,42,41)(33,39,43,44);(5,8,9,11)(6,7,10,12)(13,17,19,18)(14,23,24,22)(15,27,28,26)(16,30,31,32)(20,34,36,35)(21,29,38,37)(25,41,42,40)(33,39,43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6525067);