#7207338

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1601 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)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,32)(21,27)(22,24)(23,36)(25,30)(26,34)(29,44)(31,40)(33,42)(35,41)(37,38)(39,45)(43,46)(47,48);(1,3,6,2)(4,8,5,7)(9,11)(10,13)(12,15)(14,16)(18,24)(25,38)(26,32)(27,36)(28,40)(39,48)(41,42)(43,44);(1,4,6,5)(2,7,3,8)(11,15)(13,16)(17,19)(18,25)(20,29)(21,30)(22,31)(23,37)(24,38)(26,39)(27,40)(28,36)(32,48)(33,45)(34,46)(35,47)(41,43)(42,44);(1,4,6,5)(2,8,3,7)(10,14)(11,15)(17,20,22,34)(18,26,24,32)(19,29,31,46)(21,35,23,33)(25,39,38,48)(27,42,36,41)(28,43,40,44)(30,47,37,45);(1,5,6,4)(2,8,3,7)(11,15)(13,16)(17,21,22,23)(18,27,24,36)(19,30,31,37)(20,33,34,35)(25,40,38,28)(26,41,32,42)(29,45,46,47)(39,43,48,44)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=efi,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7114534);