#437271

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
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 ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1060 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)(6,9)(7,11)(13,14,16,18)(15,17,19,20)(21,22,26,29,27,31,24,35)(23,32,33,30,34,36,25,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,19)(18,20)(21,23,27,34)(22,28,31,30)(24,33,26,25)(29,36,35,32);(5,7,8,11)(6,10,9,12)(15,19)(17,20)(21,24,27,26)(22,29,31,35)(23,25,34,33)(28,32,30,36);(2,4)(6,9)(10,12)(21,25,26,23,27,33,24,34)(22,30,29,36,31,28,35,32)

Polycyclic presentation:

2^9|a^2=efg,b^2=i,c^2=f,d^2=efi,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei

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, 344467);