#8821912

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

4212 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)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(33,34,38,37)(35,39,43,42)(36,44,46,40)(41,48,45,47);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(34,37)(39,42)(40,44)(47,48);(2,6)(3,10)(8,12)(11,16)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30)(33,35,38,43)(34,39,37,42)(36,45,46,41)(40,48,44,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,36,38,46)(34,40,37,44)(35,41,43,45)(39,47,42,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,37,38,34)(35,42,43,39)(36,44,46,40)(41,48,45,47)

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=g,e^2=i,f^2=gi,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, 8729108);