#436664

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	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1522 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:

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

Polycyclic presentation:

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

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, 343860);