#414316

order	512
nilpotency class	3
isoclinism	isoclinic class of #395154
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

1120 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=e,d^2=gh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321512);