#417342

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

1104 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)(13,14)(15,17)(16,18)(19,20)(21,22,27,30)(23,25,29,26)(24,32,33,31)(28,34,36,35);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,16,19)(14,17,18,20)(21,23,25,30,27,29,26,22)(24,32,34,28,33,31,35,36);(6,9)(10,12)(15,19)(17,20)(21,24,25,34,27,33,26,35)(22,28,23,31,30,36,29,32);(5,7,8,11)(6,10,9,12)(14,18)(17,20)(24,33)(28,36)(31,32)(34,35)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=fi,d^2=ghi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 324538);