#416428

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

1060 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,17)(18,20)(21,22,26,28)(23,29,30,36)(24,25,32,33)(27,34,35,31);(1,2)(3,4)(6,9)(10,12)(21,23,25,31,26,30,33,34)(22,27,24,29,28,35,32,36);(2,4)(5,7,8,11)(6,10,9,12)(14,17)(18,19)(21,24,25,28,26,32,33,22)(23,29,31,35,30,36,34,27);(13,15,16,20)(14,18,17,19)(23,30)(27,35)(29,36)(31,34)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=e,d^2=h,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, 323624);