#7603456

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1584 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)(4,8)(5,6)(9,10,12,13)(11,15,16,14)(17,18)(19,26)(20,34)(21,36)(22,30)(23,29)(24,28)(25,33)(27,40)(31,44)(32,42)(35,41)(37,43)(38,46)(39,45)(47,48);(2,6)(7,8)(10,13)(11,16)(17,19)(18,25)(20,24)(21,35)(22,32)(23,33)(26,29)(27,44)(28,42)(30,34)(31,38)(36,43)(37,39)(40,47)(41,45)(46,48);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,22,33,23,32,24,19)(18,26,28,42,29,25,30,34)(21,37,38,31,39,35,40,47)(27,41,45,44,46,43,36,48);(1,4,5,3)(2,8,6,7)(10,13)(14,15)(17,21,24,40,23,39,22,38)(18,27,30,36,29,46,28,45)(19,31,32,37,33,47,20,35)(25,41,42,48,26,43,34,44);(2,6)(7,8)(10,13)(14,15)(17,22,23,24)(18,28,29,30)(19,32,33,20)(21,38,39,40)(25,42,26,34)(27,45,46,36)(31,37,47,35)(41,48,43,44)

Polycyclic presentation:

2^9|a^2=gh,c^2=fhi,d^2=fgi,e^2=i,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,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7510652);