#7368893

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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)(3,7)(4,6)(5,8)(17,18,23,27,22,24,21,28)(19,29,31,25,32,42,33,38)(20,34,35,40,36,26,37,41)(30,44,47,48,46,43,45,39);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(25,38)(26,34)(27,28)(29,42)(39,48)(40,41)(43,44);(9,11,12,16)(10,13,14,15)(17,19,22,32)(18,25,24,38)(20,30,36,46)(21,31,23,33)(26,39,34,48)(27,29,28,42)(35,45,37,47)(40,44,41,43);(3,8)(5,7)(10,14)(11,16)(17,20)(18,26)(19,30)(21,37)(22,36)(23,35)(24,34)(25,39)(27,41)(28,40)(29,43)(31,47)(32,46)(33,45)(38,48)(42,44);(2,6)(5,7)(10,14)(13,15)(17,21,22,23)(18,27,24,28)(19,31,32,33)(20,35,36,37)(25,29,38,42)(26,40,34,41)(30,45,46,47)(39,44,48,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,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, 7276089);