#8497773

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

4225 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=gh,f^2=hi,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, 8404969);