#416845

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

960 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)(5,6,8,10)(7,9,11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,29)(24,30)(25,31)(26,28)(27,32)(33,36)(34,35);(5,7)(6,9)(8,11)(10,12)(14,18)(19,20)(21,23,29,22,26,31,25,28)(24,30,35,27,33,36,32,34);(2,4)(7,11)(9,12)(13,15,17,16)(14,19,18,20)(21,24,25,32,26,33,29,35)(22,27,23,30,28,34,31,36);(13,16,17,15)(14,19,18,20)(22,28)(23,31)(27,34)(30,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=e,d^2=hi,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, 324041);