#8258614

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

4227 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8165810);