#6990078

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1616 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=g,d^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfhi,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6897274);