#7567069

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1598 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,32)(21,27)(22,28)(23,26)(25,35)(29,41)(30,40)(31,39)(33,38)(34,42)(36,43)(37,44)(45,48)(46,47);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,33)(21,30)(22,31)(23,37)(25,41)(26,44)(27,40)(28,39)(29,35)(32,38)(34,46)(36,45)(42,47)(43,48);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,22,35)(18,25,28,32)(19,29,31,33)(21,36,23,34)(24,38,39,41)(26,43,27,42)(30,46,37,45)(40,48,44,47);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21)(18,26)(20,34)(22,23)(24,39)(25,42)(27,28)(29,33)(30,37)(32,43)(35,36)(47,48);(2,6)(7,8)(10,14)(13,15)(17,21,22,23)(18,27,28,26)(19,30,31,37)(20,34,35,36)(24,40,39,44)(25,43,32,42)(29,45,33,46)(38,47,41,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=fg,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=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, 7474265);