#7603736

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

1362 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)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18,23,29)(19,24,33,42)(20,34,39,44)(21,27,22,28)(25,37,45,38)(26,32,41,31)(30,43,36,46)(35,40,47,48);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,19,23,33)(18,24,29,42)(20,35,39,47)(21,31,22,32)(25,43,45,46)(26,28,41,27)(30,37,36,38)(34,40,44,48);(1,3,5,4)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,20,22,38,23,39,21,37)(18,25,27,34,29,45,28,44)(19,30,31,35,33,36,32,47)(24,40,41,43,42,48,26,46);(1,4,5,3)(2,6,8,7)(11,16)(14,15)(17,19,21,32,23,33,22,31)(18,26,28,42,29,41,27,24)(20,36,37,35,39,30,38,47)(25,43,44,40,45,46,34,48);(2,8)(6,7)(17,21,23,22)(18,27,29,28)(19,31,33,32)(20,37,39,38)(24,26,42,41)(25,34,45,44)(30,35,36,47)(40,46,48,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fi,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^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, 7510932);