#6535914

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂×#3267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2321 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(13,14)(15,20)(16,29)(17,22)(18,23)(19,24)(21,30)(25,39)(26,34)(27,35)(28,36)(31,37)(32,38)(33,41)(40,43)(42,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,28)(14,20,24,36)(16,25,31,42)(17,27,18,26)(21,33,38,43)(22,35,23,34)(29,39,37,44)(30,41,32,40);(6,9)(10,12)(13,16)(14,21)(15,25)(17,30)(18,32)(19,31)(20,33)(22,37)(23,29)(24,38)(26,40)(27,41)(28,42)(34,39)(35,44)(36,43);(5,7)(6,10)(8,11)(9,12)(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);(7,11)(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|b^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh

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, 6443110);