#8806770

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

4202 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=hi,e^2=hi,f^2=h,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, 8713966);