#6436676

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432833
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1984 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343872);