#6558957

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1335 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6466153);