#6742001

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1539 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13)(19,28)(23,37)(29,43)(30,32)(31,36)(38,39)(44,46)(45,47);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,18,21,26)(19,29,28,43)(20,24,33,41)(22,25,35,27)(23,36,37,31)(30,44,32,46)(34,40,48,42)(38,47,39,45);(10,14)(13,15)(17,19,21,28)(18,23,26,37)(20,32,33,30)(22,36,35,31)(24,39,41,38)(25,29,27,43)(34,45,48,47)(40,46,42,44);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(17,20,21,33)(18,24,26,41)(19,30,28,32)(22,34,35,48)(23,38,37,39)(25,40,27,42)(29,44,43,46)(31,45,36,47);(1,4,5,3)(2,6,7,8)(11,16)(13,15)(18,25)(22,35)(23,29)(24,40)(26,27)(31,36)(34,48)(37,43)(38,44)(39,46)(41,42)(45,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fg,e^2=fgi,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, 6649197);