#6692518

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1580 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,12,16);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,18,22,26)(19,27,29,32)(20,25,21,24)(23,30,31,28);(9,12,13,11)(10,14,15,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31);(2,6)(5,7)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,31,32);(3,8)(5,7)(18,24)(20,21)(23,27)(25,26)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6599714);