#444731

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1046 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)(5,6)(7,9)(8,10)(11,12)(13,14,16,18)(15,17,19,20)(21,22)(23,30)(24,26)(25,27)(28,32)(29,33)(31,35)(34,36);(5,7,8,11)(6,9,10,12)(15,19)(17,20)(22,26)(23,31)(24,33)(27,29)(28,30)(32,34);(6,10)(9,12)(13,15,16,19)(14,17,18,20)(22,27)(26,29)(28,30)(35,36);(2,4)(21,23,25,32)(22,28,27,30)(24,34,33,31)(26,35,29,36)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=g,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,e^d=ei

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, 351927);