#459750

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1038 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)(3,7)(4,5)(6,8)(9,10,13,15)(11,14,12,16)(17,18,21,25)(19,28,20,23)(22,26,32,27)(24,31,29,30);(2,5)(6,7)(9,11,13,12)(10,14,15,16)(17,18,21,25)(19,29,20,24)(22,27,32,26)(23,31,28,30);(1,3)(2,6)(4,8)(5,7)(10,15)(14,16)(17,19,21,20)(18,23,25,28)(22,31,32,30)(24,27,29,26);(3,8)(6,7)(9,12,13,11)(10,16,15,14)(17,20,21,19)(18,24,25,29)(22,31,32,30)(23,27,28,26)

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=i,d^2=ef,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, 366946);