#7592816

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1284 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)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,31)(21,36)(22,38)(23,27)(25,35)(26,33)(28,41)(29,44)(30,40)(32,39)(34,48)(37,42)(43,45)(46,47);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,30)(18,24,27,40)(20,32,34,37)(21,33,29,22)(25,41,43,47)(26,44,38,36)(28,45,46,35)(31,39,48,42);(17,20,33,45,23,34,22,35)(18,25,38,48,27,43,26,31)(19,28,21,37,30,46,29,32)(24,39,44,47,40,42,36,41);(11,15)(13,16)(17,21,22,19,23,29,33,30)(18,24,26,36,27,40,38,44)(20,32,35,46,34,37,45,28)(25,42,31,47,43,39,48,41);(1,3,4,8)(2,6,5,7)(20,34)(25,43)(28,46)(31,48)(32,37)(35,45)(39,42)(41,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,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,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, 7500012);