#7180978

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996405
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1639 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi

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, 7088174);