#6442158

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,e^d=ei,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349354);