#7586076

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

1494 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)(3,6)(4,5)(7,8)(10,14)(11,12)(17,18,22,27)(19,28,32,24)(20,25,35,42)(21,36,23,26)(29,45,48,38)(30,46,47,39)(31,40,33,44)(34,43,37,41);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,22,32)(18,24,27,28)(20,29,35,48)(21,33,23,31)(25,38,42,45)(26,44,36,40)(30,37,47,34)(39,41,46,43);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,37)(22,35)(23,34)(24,38)(26,43)(27,42)(28,45)(30,31)(32,48)(33,47)(36,41)(39,44)(40,46);(9,11,13,12)(10,16,14,15)(17,20)(18,25)(19,30)(21,34)(22,35)(23,37)(24,39)(26,41)(27,42)(28,46)(29,31)(32,47)(33,48)(36,43)(38,44)(40,45);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,26,27,36)(19,31,32,33)(20,34,35,37)(24,40,28,44)(25,41,42,43)(29,47,48,30)(38,39,45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=g,e^2=gi,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, 7493272);