#8202481

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4264 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)(5,10)(6,8)(7,14)(9,15)(12,16)(17,18)(19,25)(20,30)(21,26)(22,27)(23,24)(28,29)(31,32)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(1,3,6,14)(2,7,8,11)(4,10,15,12)(5,13,16,9)(18,24)(21,27)(25,30)(28,31)(33,35,38,43)(34,39,37,41)(36,42,45,48)(40,46,44,47);(2,8)(5,16)(7,11)(9,13)(17,19,23,20)(18,25,24,30)(21,28,27,31)(22,32,26,29)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,16)(10,15)(17,20,23,19)(18,25,24,30)(21,31,27,28)(22,32,26,29)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,7,16,11)(17,21,23,27)(18,26,24,22)(19,28,20,31)(25,29,30,32)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,7,15,11)(17,22,23,26)(18,27,24,21)(19,29,20,32)(25,28,30,31)(33,37,38,34)(35,41,43,39)(36,44,45,40)(42,47,48,46)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8109677);