#8492814

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

4249 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=g,e^2=i,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, 8400010);