#6742010

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	#23963	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1637 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=fgi,e^2=fg,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, 6649206);