#8603531

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#55678	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8510727);