#7598875

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

1357 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)(11,15)(12,13)(14,16)(17,18)(19,25)(20,34)(21,39)(22,28)(23,29)(24,30)(26,37)(27,33)(31,42)(32,41)(35,40)(36,45)(38,44)(43,46)(47,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,35)(21,22)(23,32)(24,33)(26,42)(27,30)(28,39)(29,41)(31,37)(34,40)(36,47)(38,46)(43,44)(45,48);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,23,37,24,38,22,36)(18,26,28,34,30,45,29,44)(19,31,21,35,33,47,32,46)(25,40,41,42,27,43,39,48);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,14,13,15)(17,21,23,33,24,32,22,19)(18,27,28,39,30,25,29,41)(20,35,37,47,38,46,36,31)(26,43,34,48,45,40,44,42);(1,4,5,3)(2,7,8,6)(17,22,24,23)(18,28,30,29)(19,21,33,32)(20,36,38,37)(25,39,27,41)(26,34,45,44)(31,35,47,46)(40,48,43,42)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fhi,d^2=fg,e^2=fi,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, 7506071);