#427657

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

1034 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)(5,6,8,10)(7,11,12,9)(13,14)(15,18)(16,17)(19,20)(21,22,24,27)(23,29,31,26)(25,28,32,35)(30,36,33,34);(5,7,8,12)(6,9,10,11)(23,30)(25,32)(26,34)(28,35)(29,36)(31,33);(14,17)(18,20)(21,23,24,31)(22,26,27,29)(25,33,32,30)(28,36,35,34);(2,4)(7,12)(9,11)(13,15,16,19)(14,18,17,20)

Polycyclic presentation:

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

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, 334853);