#7183379

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1393 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fhi,c^2=f,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 7090575);