#459285

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
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

852 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)(8,12)(9,11)(13,14,17,16)(15,19,20,18)(21,22,25,28)(23,26,31,35)(24,27,32,36)(29,33,30,34);(5,7,9,8)(6,10,11,12)(13,15,17,20)(14,18,16,19)(21,23,25,31)(22,26,28,35)(24,29,32,30)(27,33,36,34);(6,11)(10,12)(13,16,17,14)(15,19,20,18)(23,29)(24,32)(26,33)(27,36)(30,31)(34,35);(2,4)(5,8,9,7)(6,12,11,10)(14,16)(18,19)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366481);