#7540069

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1524 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)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,31)(21,25)(22,26)(23,28)(24,33)(29,36)(30,42)(32,40)(34,39)(35,38)(37,44)(41,43)(45,47)(46,48);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(19,28)(20,32)(22,34)(23,27)(24,38)(26,39)(29,45)(31,40)(33,35)(36,47)(41,48)(43,46);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19,21,28)(18,23,25,27)(20,29,33,43)(22,30,34,44)(24,36,31,41)(26,37,39,42)(32,46,35,45)(38,48,40,47);(10,14)(13,15)(17,20,21,33)(18,24,25,31)(19,29,28,43)(22,35,34,32)(23,36,27,41)(26,40,39,38)(30,45,44,46)(37,47,42,48);(2,5)(6,8)(10,14)(13,15)(18,25)(19,28)(24,31)(26,39)(29,43)(30,44)(38,40)(45,46)

Polycyclic presentation:

2^9|b^2=gi,c^2=fg,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,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, 7447265);