#8581498

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3895 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)(3,11)(4,13)(5,7)(6,10)(8,12)(9,15)(14,16)(17,18,23,20)(19,21,24,26)(22,25,30,28)(27,29,31,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,48)(45,47);(1,3,7,12)(2,8,5,11)(4,14,15,6)(9,16,13,10)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,32)(30,31)(33,34,38,42)(35,39,46,43)(36,40,37,41)(44,47,45,48);(2,5)(8,11)(9,13)(10,16)(17,18,23,20)(19,26,24,21)(22,25,30,28)(27,32,31,29)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4,7,15)(2,9,5,13)(3,6,12,14)(8,10,11,16)(17,20,23,18)(19,26,24,21)(22,25,30,28)(27,29,31,32)(33,35,38,46)(34,39,42,43)(36,44,37,45)(40,47,41,48);(1,5,7,2)(3,11,12,8)(4,13,15,9)(6,16,14,10)(17,21,23,26)(18,24,20,19)(22,29,30,32)(25,31,28,27)(33,36,38,37)(34,40,42,41)(35,44,46,45)(39,47,43,48);(1,6)(2,10)(3,4)(5,16)(7,14)(8,9)(11,13)(12,15)(17,22,23,30)(18,25,20,28)(19,27,24,31)(21,29,26,32)(33,37,38,36)(34,41,42,40)(35,45,46,44)(39,48,43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8488694);