#6679819

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1458 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)(3,5)(4,6)(7,8)(9,10,12,13)(11,15,16,14)(17,18)(19,24)(20,31)(21,26)(22,27)(23,25)(28,30)(29,32);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,27)(19,28,26,32)(20,25,23,31)(21,30,24,29);(2,6)(5,8)(10,13)(14,15)(17,19,22,26)(18,24,27,21)(20,30,23,29)(25,32,31,28);(9,11,12,16)(10,14,13,15)(17,20,22,23)(18,25,27,31)(19,29,26,30)(21,32,24,28);(17,21,22,24)(18,26,27,19)(20,32,23,28)(25,30,31,29)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6587015);