#7526258

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1454 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)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,33)(21,36)(22,28)(23,29)(24,30)(26,35)(27,39)(31,46)(32,42)(34,43)(37,48)(38,45)(40,44)(41,47);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,32)(18,25,29,42)(20,24,35,22)(21,37,39,34)(26,28,33,30)(27,43,36,48)(31,40,47,38)(41,45,46,44);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20)(18,26)(19,22)(21,34)(23,35)(24,32)(25,30)(27,43)(28,42)(29,33)(31,38)(36,48)(37,39)(40,47)(41,45)(44,46);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,21,23,39)(18,27,29,36)(19,31,32,47)(20,34,35,37)(22,38,24,40)(25,41,42,46)(26,43,33,48)(28,44,30,45);(2,8)(6,7)(17,22,23,24)(18,28,29,30)(19,20,32,35)(21,38,39,40)(25,33,42,26)(27,44,36,45)(31,34,47,37)(41,48,46,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7433454);