#432872

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340068);