#417210

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=eh,c^2=fg,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^c=di

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, 324406);