#8535229

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

4274 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(18,24)(19,20)(21,29)(30,32)(34,38)(39,45)(40,43)(46,47);(3,4)(7,10)(11,12)(15,16)(17,18)(19,25)(20,22)(21,26)(23,24)(27,32)(28,30)(29,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,45)(43,44)(47,48);(1,3,6,4)(2,7,9,10)(5,12,14,11)(8,16,13,15)(17,19,23,20)(18,25,24,22)(21,28,29,27)(26,30,31,32)(33,35,37,42)(34,39,38,45)(36,41,44,48)(40,46,43,47);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20,23,19)(18,25,24,22)(21,28,29,27)(26,32,31,30)(33,34)(35,39)(36,43)(37,38)(40,44)(41,47)(42,45)(46,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,21,23,29)(18,26,24,31)(19,27,20,28)(22,30,25,32)(33,35)(34,39)(36,41)(37,42)(38,45)(40,46)(43,47)(44,48);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,22,23,25)(18,20,24,19)(21,30,29,32)(26,28,31,27)(33,36,37,44)(34,40,38,43)(35,41,42,48)(39,46,45,47)

Polycyclic presentation:

2^9|c^2=gi,d^2=h,e^2=hi,f^2=g,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, 8442425);