#7604637

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1287 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=h,d^2=g,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, 7511833);