#472750

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
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

842 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)(5,6,9,11)(7,10,8,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,35)(33,34);(1,2,4,3)(5,7,9,8)(6,10,11,12)(13,15,16,19)(14,17,18,20)(21,23,26,31,25,24,32,33)(22,27,29,34,28,30,35,36);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(21,24)(22,27)(23,25)(26,31)(28,30)(29,36)(32,33)(34,35);(1,3,4,2)(14,18)(17,20)(23,24)(27,30)(31,33)(34,36)

Polycyclic presentation:

2^9|a^2=efi,b^2=fgh,c^2=efi,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,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, 379946);