#7605738

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1305 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fi,c^2=fhi,d^2=fg,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,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7512934);