#421673

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

1030 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efi,c^2=e,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,h^d=hi

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, 328869);