#6460513

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1619 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,26,29,23)(20,31,25,21)(27,28,32,30);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,29)(18,23,24,26)(20,32,25,27)(21,30,31,28);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20,21,18,22,25,31,24)(19,27,30,26,29,32,28,23);(1,4,5,3)(2,6,8,7)(11,15)(13,16)(17,21,22,31)(18,20,24,25)(19,28,29,30)(23,27,26,32);(19,29)(23,26)(27,32)(28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367709);