#8756987

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

4117 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)(3,5,10,8)(4,12,14,9)(11,15,16,13)(19,25)(22,26)(27,31)(30,32)(33,34,39,44)(35,40,47,46)(36,43,38,41)(37,42,45,48);(2,7)(5,8)(9,12)(13,15)(17,18)(19,26)(20,28)(21,24)(22,25)(23,29)(27,30)(31,32)(33,35)(34,40)(36,37)(38,45)(39,47)(41,48)(42,43)(44,46);(2,7)(3,10)(9,12)(11,16)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,36,39,38)(34,41,44,43)(35,45,47,37)(40,42,46,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,37,39,45)(34,42,44,48)(35,36,47,38)(40,43,46,41);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,38,39,36)(34,43,44,41)(35,45,47,37)(40,42,46,48);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,34)(35,46)(36,41)(37,42)(38,43)(39,44)(40,47)(45,48)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664183);