#477246

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	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

960 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)(3,6)(4,7)(5,8)(9,10,12,13)(11,15,16,14)(17,18,23,27)(19,24,20,30)(21,26,22,32)(25,29,28,31);(1,3)(2,5)(4,8)(6,7)(10,13)(14,15)(17,18,23,27)(19,28,20,25)(21,32,22,26)(24,31,30,29);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,14,13,15)(17,19)(18,24)(20,23)(21,31)(22,29)(25,26)(27,30)(28,32);(3,8)(5,6)(11,16)(14,15)(17,20)(18,25)(19,23)(21,31)(22,29)(24,26)(27,28)(30,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=f,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, 384442);