#8284465

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

4268 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8191661);