#8587654

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

4033 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8494850);