#7171760

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1253 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7078956);