#437467

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

962 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)(7,10)(8,11)(9,12)(13,14,16,18)(15,17,19,20)(21,22,24,29,25,27,32,26)(23,28,30,35,31,34,36,33);(5,7,9,8)(6,10,12,11)(13,15)(14,17)(16,19)(18,20)(22,26)(23,30)(24,32)(27,29)(31,36)(33,35);(5,7,9,8)(6,11,12,10)(15,19)(17,20)(22,27)(26,29)(28,34)(33,35);(2,4)(5,8,9,7)(6,11,12,10)(21,23,24,30,25,31,32,36)(22,28,29,35,27,34,26,33)

Polycyclic presentation:

2^9|a^2=eg,b^2=fi,c^2=fi,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, 344663);