#8464214

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

4153 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)(3,10)(4,7)(5,12)(6,8)(9,14)(11,15)(13,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,30)(31,32)(34,40)(35,43)(36,45)(42,48);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(19,30)(20,31)(24,29)(25,32)(33,34,39,40)(35,41,43,47)(36,46,45,38)(37,42,44,48);(1,2,7,4)(3,10,6,8)(5,9,14,12)(11,16,13,15)(17,19)(18,24)(20,27)(21,31)(22,32)(23,30)(25,26)(28,29)(33,35)(34,41)(36,44)(37,45)(38,42)(39,43)(40,47)(46,48);(1,4,7,2)(3,8,6,10)(5,12,14,9)(11,15,13,16)(17,20,23,31)(18,25,28,32)(19,27,30,21)(22,24,26,29)(33,36,39,45)(34,38,40,46)(35,44,43,37)(41,42,47,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,21,23,27)(18,26,28,22)(19,31,30,20)(24,25,29,32)(33,37,39,44)(34,42,40,48)(35,45,43,36)(38,47,46,41);(1,6,7,3)(2,8,4,10)(5,13,14,11)(9,15,12,16)(17,22,23,26)(18,27,28,21)(19,32,30,25)(20,29,31,24)(33,38)(34,36)(35,42)(37,47)(39,46)(40,45)(41,44)(43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8371410);