#6889351

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6796547);