#478188

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

854 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)(5,6,10,7)(8,12,9,11)(13,14,16,19)(15,18,20,17)(21,22)(23,26)(24,27)(25,28)(29,36)(30,34)(31,33)(32,35);(5,7,10,6)(8,12,9,11)(13,15,16,20)(14,17,19,18)(21,23)(22,26)(24,30)(25,29)(27,34)(28,36)(31,32)(33,35);(5,8,10,9)(6,11,7,12)(13,15)(14,18)(16,20)(17,19)(22,27)(23,29)(25,32)(26,33)(30,31)(34,36);(2,4)(5,9,10,8)(6,11,7,12)(15,20)(17,18)(23,30)(26,34)(29,31)(33,36)

Polycyclic presentation:

2^9|a^2=efi,b^2=efi,c^2=fi,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385384);