#433309

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1002 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=fi,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=de,h^d=hi

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, 340505);