#7587006

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

1517 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)(19,24)(20,26)(21,27)(22,28)(23,29)(25,35)(30,43)(31,39)(32,40)(33,41)(34,44)(36,42)(37,45)(38,46)(47,48);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,16,12,15)(17,19,22,33)(18,24,28,41)(20,30,35,46)(21,32,23,37)(25,38,26,43)(27,40,29,45)(31,36,47,34)(39,42,48,44);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,30)(21,36)(22,35)(23,34)(24,38)(26,28)(27,44)(29,42)(31,37)(32,47)(33,46)(39,40)(41,43)(45,48);(9,11,13,12)(10,16,14,15)(17,20)(18,26)(19,31)(21,34)(22,35)(23,36)(24,39)(25,28)(27,44)(29,42)(30,37)(32,46)(33,47)(38,40)(41,48)(43,45);(2,7)(6,8)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,27,28,29)(19,32,33,37)(20,34,35,36)(24,40,41,45)(25,42,26,44)(30,31,46,47)(38,48,43,39)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=gi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,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, 7494202);