#415439

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
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

1012 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)(13,14,16,17)(15,19,20,18)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(2,4)(5,7,9,8)(6,10,11,12)(14,17)(18,19)(21,23)(22,26)(24,31)(25,30)(27,29)(28,34)(32,33)(35,36);(2,4)(5,8,9,7)(6,10,11,12)(14,17)(18,19)(23,30)(25,33)(26,34)(28,36)(29,35)(31,32);(13,15,16,20)(14,18,17,19)(22,27)(26,29)(28,36)(34,35)

Polycyclic presentation:

2^9|a^2=efh,b^2=efi,c^2=efi,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,g^c=gi

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, 322635);