#398699

order	512
nilpotency class	3
isoclinism	isoclinic class of #393909
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

1152 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)(7,12)(8,10)(9,11)(13,14,16,18)(15,19,20,17)(21,22,24,26)(23,29,30,36)(25,33,32,28)(27,34,35,31);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,16,20)(14,17,18,19)(22,26)(27,35)(28,33)(29,36);(2,4)(5,8,9,7)(6,10,11,12)(14,18)(17,19)(22,26)(27,35)(28,33)(29,36);(21,23,24,30)(22,27,26,35)(25,34,32,31)(28,36,33,29)

Polycyclic presentation:

2^9|a^2=f,b^2=ei,c^2=ef,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,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, 305895);