#8551596

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

3952 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8458792);