#8535327

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

4191 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=h,e^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, 8442523);