#7302570

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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	#45091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1656 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,e^c=ei,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, 7209766);