#8745862

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
prime radical	#47542	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=i,d^2=hi,e^2=g,f^2=gh,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, 8653058);