#415345

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

1212 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=ef,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=dh,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, 322541);