#8928678

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

4274 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8835874);