#459878

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1006 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=e,d^2=efi,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, 367074);