#7605231

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=fhi,d^2=gi,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, 7512427);