#7602679

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

1365 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)(9,10,12,14)(11,15,16,13)(17,18,24,30)(19,25,33,27)(20,34,36,45)(21,39,32,42)(22,29,23,28)(26,38,44,37)(31,47,48,43)(35,46,40,41);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,24,33)(18,25,30,27)(20,35,36,40)(21,22,32,23)(26,43,44,47)(28,39,29,42)(31,37,48,38)(34,46,45,41);(1,3,4,7)(2,6,5,8)(17,20,22,37,24,36,23,38)(18,26,28,34,30,44,29,45)(19,31,21,40,33,48,32,35)(25,41,42,47,27,46,39,43);(11,16)(13,15)(17,21,23,19,24,32,22,33)(18,27,29,42,30,25,28,39)(20,35,38,48,36,40,37,31)(26,41,45,43,44,46,34,47);(2,5)(6,8)(10,14)(13,15)(20,36)(26,44)(31,48)(34,45)(35,40)(37,38)(41,46)(43,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=fhi,d^2=gi,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=cgi,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, 7509875);