#8492789

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

4093 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)(3,11)(4,5)(6,8)(7,10)(9,15)(12,14)(13,16)(17,18,23,27)(19,24,28,32)(20,22,26,31)(21,30,29,25)(33,34)(35,43)(36,42)(37,47)(38,40)(39,41)(44,45)(46,48);(1,3,7,6)(2,8,10,11)(4,12,15,13)(5,16,9,14)(19,28)(21,29)(24,32)(25,30)(33,35,40,39)(34,41,38,43)(36,44,47,46)(37,45,42,48);(1,4)(2,9)(3,12)(5,10)(6,13)(7,15)(8,14)(11,16)(17,19,23,28)(18,24,27,32)(20,29,26,21)(22,25,31,30)(33,36,40,47)(34,42,38,37)(35,44,39,46)(41,48,43,45);(1,4)(2,9)(3,13)(5,10)(6,12)(7,15)(8,16)(11,14)(17,20)(18,22)(19,29)(21,28)(23,26)(24,25)(27,31)(30,32)(33,37)(34,36)(35,45)(38,47)(39,48)(40,42)(41,44)(43,46);(1,5,7,9)(2,4,10,15)(3,14,6,16)(8,13,11,12)(17,21,23,29)(18,25,27,30)(19,20,28,26)(22,24,31,32)(33,38,40,34)(35,41,39,43)(36,37,47,42)(44,48,46,45);(1,6,7,3)(2,8,10,11)(4,13,15,12)(5,16,9,14)(17,22)(18,26)(19,25)(20,27)(21,24)(23,31)(28,30)(29,32)(33,39,40,35)(34,41,38,43)(36,46,47,44)(37,45,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8399985);