#6536639

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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1804 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)(3,6)(4,5)(7,8)(17,18,23,26)(19,20,28,31)(21,24,32,29)(22,27,30,25);(9,10)(11,14)(12,13)(15,16)(17,19,22,29,23,28,30,24)(18,21,25,20,26,32,27,31);(2,5)(6,8)(10,13)(14,16)(17,20)(18,19)(21,30)(22,32)(23,31)(24,27)(25,29)(26,28);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(17,21,23,32)(18,24,26,29)(19,27,28,25)(20,30,31,22);(11,15)(14,16)(19,28)(20,31)(21,32)(24,29)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ei,e^c=ei,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, 6443835);