#8993164

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

4112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8900360);