#6546574

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1709 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^c=ei,e^d=ehi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313494, #72313531, #72314044
maximal normal subgroup of	#72313494, #72313531, #72314044

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6453770);