#411502

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

1062 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=eh,c^2=e,d^2=g,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, 318698);