#415565

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1216 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)(13,14,16,18)(15,19,20,17)(21,22,25,27)(23,29,30,36)(24,32,34,28)(26,33,35,31);(5,6,8,10)(7,11,12,9)(21,23,25,30)(22,26,27,35)(24,33,34,31)(28,36,32,29);(2,4)(5,7,8,12)(6,9,10,11)(23,30)(26,35)(29,36)(31,33);(13,15,16,20)(14,17,18,19)(22,27)(26,35)(28,32)(29,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322761);