#8680007

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

4176 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)(3,10)(4,13)(5,7)(6,8)(9,14)(11,15)(12,16)(18,24)(20,25)(26,32)(28,30);(1,3,7,6)(2,8,5,10)(4,11,14,12)(9,15,13,16)(17,18,23,24)(19,25,21,20)(22,26,31,32)(27,30,29,28)(33,34)(35,40)(36,39)(37,41)(38,42)(43,44)(45,47)(46,48);(1,4)(2,9)(3,11)(5,13)(6,12)(7,14)(8,15)(10,16)(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,30,32,28)(33,35)(34,37)(36,43)(38,41)(39,46)(40,42)(44,47)(45,48);(1,4)(2,9)(3,12)(5,13)(6,11)(7,14)(8,16)(10,15)(17,20)(18,19)(21,24)(22,30)(23,25)(26,29)(27,32)(28,31)(33,36,38,45)(34,39,42,47)(35,43,41,48)(37,46,40,44);(1,5,7,2)(3,10,6,8)(4,13,14,9)(11,16,12,15)(17,21,23,19)(18,20,24,25)(22,27,31,29)(26,30,32,28)(33,35)(34,40)(36,43)(37,42)(38,41)(39,44)(45,48)(46,47);(1,6,7,3)(2,8,5,10)(4,12,14,11)(9,15,13,16)(17,22,23,31)(18,26,24,32)(19,27,21,29)(20,28,25,30)(33,37,38,40)(34,41,42,35)(36,44,45,46)(39,43,47,48)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8587203);