#8202291

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3855 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,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, 8109487);