#7090137

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1416 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,29)(19,30,33,25)(20,26,36,44)(21,28,22,27)(24,37,40,32)(31,42,47,41)(34,39,46,48)(35,45,38,43);(9,10)(11,15)(12,14)(13,16)(17,19,23,33)(18,24,29,40)(20,34,36,46)(21,32,22,37)(25,27,30,28)(26,42,44,41)(31,35,47,38)(39,43,48,45);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,31)(21,37)(22,32)(23,33)(24,28)(26,41)(27,40)(29,30)(34,38)(35,46)(36,47)(39,45)(42,44)(43,48);(11,16)(13,15)(17,20,23,36)(18,26,29,44)(19,31,33,47)(21,35,22,38)(24,39,40,48)(25,41,30,42)(27,43,28,45)(32,34,37,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,27,29,28)(19,32,33,37)(20,35,36,38)(24,30,40,25)(26,43,44,45)(31,34,47,46)(39,42,48,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6997333);