#6468302

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1476 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)(9,10,13,16)(11,14,12,15)(17,18)(19,21)(20,24)(22,27)(23,25)(26,29)(28,31)(30,32);(11,12)(14,15)(17,19,21,18,23,27,22,25)(20,28,29,32,30,26,31,24);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,20)(18,24)(19,26)(21,29)(22,31)(23,30)(25,32)(27,28);(9,12,13,11)(10,15,16,14)(17,21,23,22)(18,19,25,27)(20,29,30,31)(24,26,32,28);(1,3,4,7)(2,6,5,8)(20,30)(24,32)(26,28)(29,31)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=hi,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375498);