#474150

order	512
nilpotency class	3
isoclinism	isoclinic class of #453960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1018 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=egh,c^2=ef,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de

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, 381346);