#7511078

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1262 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)(3,4)(5,7)(6,8)(9,10,12,13)(11,15,16,14)(17,18,22,25)(19,23,29,37)(20,24,31,40)(21,26,35,42)(27,36,45,47)(28,38,33,43)(30,39,46,48)(32,41,34,44);(1,3)(2,6)(4,7)(5,8)(17,19,22,29)(18,23,25,37)(20,30,31,46)(21,33,35,28)(24,39,40,48)(26,43,42,38)(27,34,45,32)(36,44,47,41);(10,13)(14,15)(17,20,22,31)(18,24,25,40)(19,27,29,45)(21,34,35,32)(23,36,37,47)(26,44,42,41)(28,46,33,30)(38,48,43,39);(2,7)(3,8)(9,11,12,16)(10,14,13,15)(18,25)(23,37)(24,40)(26,42)(36,47)(38,43)(39,48)(41,44);(1,4,5,6)(2,3,7,8)(18,25)(20,31)(23,37)(26,42)(27,45)(30,46)(32,34)(38,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi

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, 7418274);