#7090142

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=i,e^2=gi,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, 6997338);