#7101020

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1403 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ghi,d^2=g,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,f^b=fi,d^c=dh,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, 7008216);