#7589569

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1546 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=fg,e^2=fgi,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, 7496765);