#7593089

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1359 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fi,c^2=gi,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7500285);