#431983

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=fi,c^2=ef,d^2=efi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339179);