#6713554

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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)(4,6)(5,8)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,37)(28,31)(29,43)(30,38)(32,39)(33,41)(34,40)(35,44)(36,42)(45,46)(47,48);(1,3)(2,5)(4,8)(6,7)(17,19)(18,23)(20,32)(21,29)(22,31)(24,39)(25,28)(26,43)(27,33)(30,36)(34,47)(35,46)(37,41)(38,42)(40,48)(44,45);(10,13)(14,16)(17,20,22,34)(18,24,25,40)(19,27,31,46)(21,35,36,33)(23,37,28,45)(26,44,42,41)(29,32,30,47)(38,48,43,39);(2,6)(5,7)(17,18)(19,28)(20,24)(21,26)(22,25)(23,31)(27,45)(29,38)(30,43)(32,48)(33,41)(34,40)(35,44)(36,42)(37,46)(39,47);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(18,25)(19,29)(21,36)(23,38)(24,40)(27,32)(28,43)(30,31)(33,35)(37,48)(39,45)(46,47)

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6620750);