#417155

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×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

1480 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=fg,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 324351);