#7055144

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	#42910	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1739 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:

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

Polycyclic presentation:

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

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, 6962340);