#416115

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1110 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)(5,6,9,11)(7,12,8,10)(13,14,17,18)(15,19,16,20)(21,22)(23,27)(24,29)(25,28)(26,30)(31,34)(32,36)(33,35);(5,7,9,8)(6,10,11,12)(14,18)(19,20)(21,23)(22,27)(24,32)(25,31)(26,33)(28,34)(29,36)(30,35);(2,4)(5,8,9,7)(6,10,11,12)(13,15,17,16)(14,19,18,20)(22,28)(23,31)(24,25)(29,30)(32,33)(34,36);(13,16,17,15)(14,19,18,20)(21,24,26,25)(22,28,30,29)(23,32,33,31)(27,34,35,36)

Polycyclic presentation:

2^9|a^2=e,b^2=ehi,c^2=e,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi

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, 323311);