#6537107

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1713 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=f,d^2=hi,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=ehi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6444303);