#404595

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 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)(8,9)(11,12)(21,22,27,30)(23,31,33,32)(24,26,29,25)(28,35,36,34);(6,9)(10,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,35,27,33,25,34)(22,28,24,32,30,36,29,31);(5,7,8,11)(6,10,9,12)(14,17)(18,20)(21,24,26,30,27,29,25,22)(23,32,35,36,33,31,34,28);(2,4)(13,15,16,19)(14,18,17,20)(23,33)(28,36)(31,32)(34,35)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ei,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=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, 311791);