#7593624

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

1233 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)(9,10,12,14)(11,13,15,16)(17,18,23,29)(19,27,33,25)(20,34,38,44)(21,40,32,42)(22,30,24,28)(26,37,45,39)(31,43,47,48)(35,41,36,46);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,33)(18,25,29,27)(20,35,38,36)(21,22,32,24)(26,43,45,48)(28,42,30,40)(31,39,47,37)(34,46,44,41);(1,3,5,4)(2,7,6,8)(17,20,22,37,23,38,24,39)(18,26,28,44,29,45,30,34)(19,31,21,35,33,47,32,36)(25,41,40,43,27,46,42,48);(11,15)(13,16)(17,21,24,19,23,32,22,33)(18,27,30,40,29,25,28,42)(20,36,39,47,38,35,37,31)(26,41,34,48,45,46,44,43);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,28,29,30)(19,32,33,21)(20,37,38,39)(25,42,27,40)(26,44,45,34)(31,36,47,35)(41,48,46,43)

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=fg,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=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=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, 7500820);