#8603517

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	#38283	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4134 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:

(2,6)(7,9)(8,12)(14,15)(17,18,23,25)(19,27,29,24)(20,28,31,26)(21,30,32,22)(33,34)(35,41)(36,46)(37,42)(38,40)(39,43)(44,48)(45,47);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19,23,29)(18,24,25,27)(20,30,31,22)(21,26,32,28)(33,35,40,45)(34,41,38,47)(36,43,37,44)(39,42,48,46);(3,10)(7,9)(11,16)(14,15)(18,25)(21,32)(24,27)(26,28)(33,36,40,37)(34,42,38,46)(35,44,45,43)(39,47,48,41);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,20,23,31)(18,26,25,28)(19,22,29,30)(21,27,32,24)(33,37,40,36)(34,42,38,46)(35,43,45,44)(39,47,48,41);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,21)(18,22)(19,28)(20,27)(23,32)(24,31)(25,30)(26,29)(33,38,40,34)(35,41,45,47)(36,42,37,46)(39,44,48,43);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22)(18,21)(19,20)(23,30)(24,26)(25,32)(27,28)(29,31)(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=h,c^2=i,d^2=h,e^2=gh,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, 8510713);