#438786

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1066 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)(6,7)(9,10,13,12)(11,15,16,14)(17,18,21,24)(19,22,28,32)(20,23,25,29)(26,30,27,31);(5,6)(7,8)(9,11,13,16)(10,14,12,15)(17,19,25,26,21,28,20,27)(18,22,29,30,24,32,23,31);(5,6,8,7)(19,26)(20,25)(22,30)(23,29)(27,28)(31,32);(2,4)(5,7,8,6)(9,12,13,10)(11,14,16,15)

Polycyclic presentation:

2^9|a^2=efi,b^2=egi,c^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^c=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, 345982);