#410688

order	512
nilpotency class	3
isoclinism	isoclinic class of #395154
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

1198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^c=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, 317884);