#6476368

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469956
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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1836 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)(17,18,22,23,21,26,20,25)(19,24,28,30,27,32,29,31);(9,11)(10,13)(12,15)(14,16)(18,23)(19,27)(20,22)(24,31)(25,26)(30,32);(2,5)(6,8)(11,15)(13,16)(17,19)(18,24)(20,29)(21,27)(22,28)(23,30)(25,31)(26,32);(11,15)(13,16)(17,20,21,22)(18,25,26,23)(19,28,27,29)(24,30,32,31);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(18,26)(23,25)(24,32)(30,31)

Polycyclic presentation:

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

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, 6383564);