#8127308

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

4151 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)(11,12)(15,16)(19,26)(21,29)(23,24)(25,28)(34,37)(35,41)(36,44)(47,48);(1,3)(2,4)(5,11)(6,7)(8,12)(9,10)(13,16)(14,15)(17,18)(19,24)(20,22)(21,25)(23,26)(27,31)(28,29)(30,32)(33,34,38,37)(35,39,41,43)(36,45,44,40)(42,48,46,47);(1,4,6,10)(2,7,9,3)(5,12,14,16)(8,15,13,11)(17,19,22,26)(18,23,20,24)(21,27,29,32)(25,30,28,31)(33,35,38,41)(34,39,37,43)(36,46,44,42)(40,48,45,47);(1,4)(2,3)(5,12)(6,10)(7,9)(8,11)(13,15)(14,16)(17,20,22,18)(19,23,26,24)(21,28,29,25)(27,30,32,31)(33,36,38,44)(34,40,37,45)(35,42,41,46)(39,47,43,48);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,14)(15,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,30)(29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,5,6,14)(2,8,9,13)(3,11,7,15)(4,12,10,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37,38,34)(35,43,41,39)(36,45,44,40)(42,48,46,47)

Polycyclic presentation:

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

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, 8034504);