#8720711

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

4032 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)(3,6)(4,14)(5,16)(7,11)(8,13)(9,15)(10,12)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,28,31,32)(33,34)(35,37)(36,45)(38,40)(39,42)(41,44)(43,47)(46,48);(1,3)(2,8)(4,5)(6,11)(7,13)(9,16)(10,14)(12,15)(17,19,21,26)(18,22,24,29)(20,25,27,31)(23,28,30,32)(33,35)(34,37)(36,46)(38,43)(39,44)(40,47)(41,42)(45,48);(1,4,7,15)(2,9,11,14)(3,5,13,12)(6,10,8,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,36,39,38)(34,40,42,45)(35,43,44,46)(37,48,41,47);(1,4)(2,9)(3,12)(5,13)(6,16)(7,15)(8,10)(11,14)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,31)(30,32)(33,37,39,41)(34,35,42,44)(36,47,38,48)(40,46,45,43);(1,5,7,12)(2,10,11,16)(3,4,13,15)(6,9,8,14)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,35)(34,41)(36,43)(37,42)(38,46)(39,44)(40,47)(45,48);(1,6,7,8)(2,3,11,13)(4,10,15,16)(5,14,12,9)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,28,29,32)(33,38,39,36)(34,40,42,45)(35,43,44,46)(37,47,41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627907);