#8749711

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

3967 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)(3,5)(4,9)(6,10)(7,11)(8,14)(12,15)(13,16)(17,18,23,24)(19,28,22,25)(20,26,30,32)(21,29,31,27)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3)(2,8)(4,12)(5,11)(6,13)(7,14)(9,10)(15,16)(18,24)(20,30)(25,28)(27,29)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,4,7,13)(2,9,11,16)(3,12,14,6)(5,15,8,10)(17,19,23,22)(18,25,24,28)(20,27,30,29)(21,26,31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,4)(2,9)(3,12)(5,15)(6,14)(7,13)(8,10)(11,16)(17,20,23,30)(18,26,24,32)(19,29,22,27)(21,28,31,25)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,5,7,8)(2,3,11,14)(4,15,13,10)(6,9,12,16)(17,21)(18,27)(19,26)(20,28)(22,32)(23,31)(24,29)(25,30)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,6,7,12)(2,10,11,15)(3,13,14,4)(5,16,8,9)(17,22,23,19)(18,25,24,28)(20,29,30,27)(21,26,31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8656907);