#7535494

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,23,28)(19,24,33,41)(20,25,35,43)(21,26,22,27)(29,37,46,48)(30,38,45,47)(31,39,32,40)(34,42,36,44);(2,6)(7,8)(11,16)(13,15)(17,19,23,33)(18,24,28,41)(20,29,35,46)(21,32,22,31)(25,37,43,48)(26,40,27,39)(30,36,45,34)(38,44,47,42);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,36)(22,34)(23,35)(24,37)(26,44)(27,42)(28,43)(30,31)(32,45)(33,46)(38,39)(40,47)(41,48);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,25)(19,30)(21,34)(22,36)(23,35)(24,38)(26,42)(27,44)(28,43)(29,31)(32,46)(33,45)(37,39)(40,48)(41,47);(1,4,5,3)(2,7,6,8)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,34,35,36)(24,39,41,40)(25,42,43,44)(29,45,46,30)(37,47,48,38)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7442690);