#425928

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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

989 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333124);