#8772974

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

4134 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=ghi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8680170);