#7489966

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	#53078	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1605 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)(4,5)(7,8)(17,18,24,37,23,29,22,28)(19,25,38,47,32,42,31,41)(20,27,33,26,21,35,34,43)(30,44,45,39,36,40,46,48);(2,5)(3,7)(10,14)(11,12)(17,19)(18,25)(20,30)(21,36)(22,31)(23,32)(24,38)(26,39)(27,44)(28,41)(29,42)(33,45)(34,46)(35,40)(37,47)(43,48);(9,10)(11,16)(12,15)(13,14)(17,20,23,21)(18,26,29,43)(19,30,32,36)(22,33,24,34)(25,39,42,48)(27,28,35,37)(31,45,38,46)(40,47,44,41);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,21,23,20)(18,27,29,35)(19,30,32,36)(22,33,24,34)(25,40,42,44)(26,28,43,37)(31,46,38,45)(39,47,48,41);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,22,23,24)(18,28,29,37)(19,31,32,38)(20,33,21,34)(25,41,42,47)(26,35,43,27)(30,45,36,46)(39,40,48,44)

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=fhi,e^2=hi,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, 7397162);