#6544175

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2828 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)(21,22,24,28,25,26,33,32)(23,29,31,36,30,27,34,35);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,22)(23,27)(24,32)(25,26)(28,33)(29,30)(31,36)(34,35);(2,4)(6,10)(11,17)(12,18)(19,20)(22,26)(23,30)(28,32)(31,34);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,23,25,30)(22,27,26,29)(24,31,33,34)(28,35,32,36);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(23,30)(27,29)(31,34)(35,36)

Polycyclic presentation:

2^9|a^2=fgi,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,f^b=fi,d^c=di,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, 6451371);