#457676

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364872);