#476346

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23390	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1077 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383542);