#7586875

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1520 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)(3,8)(4,6)(5,7)(10,14)(11,12)(17,18,22,27)(19,24,32,41)(20,34,36,25)(21,26,23,28)(29,45,47,38)(30,48,46,39)(31,40,33,42)(35,44,37,43);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,16,12,15)(17,19)(18,24)(20,29)(21,33)(22,32)(23,31)(25,38)(26,42)(27,41)(28,40)(30,35)(34,45)(36,47)(37,46)(39,43)(44,48);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,37)(22,36)(23,35)(24,38)(26,44)(27,34)(28,43)(30,31)(32,47)(33,46)(39,40)(41,45)(42,48);(9,11,13,12)(10,16,14,15)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,39)(26,43)(27,34)(28,44)(29,31)(32,46)(33,47)(38,40)(41,48)(42,45);(2,7)(6,8)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,26,27,28)(19,31,32,33)(20,35,36,37)(24,40,41,42)(25,43,34,44)(29,46,47,30)(38,48,45,39)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=gi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,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, 7494071);