#198479

order	512
nilpotency class	3
isoclinism	isoclinic class of #198171
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3186	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1467 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,g^c=ghi

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, 105675);