#8746269

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
prime radical	#45063	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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)(3,7)(4,9)(5,10)(6,11)(8,12)(13,15)(14,16)(17,18,21,24)(19,25,26,22)(20,23,30,29)(27,32,28,31)(33,34,39,40)(35,43,37,41)(36,42,46,38)(44,48,47,45);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,15)(10,16)(19,26)(22,25)(27,28)(31,32)(34,40)(36,46)(41,43)(44,47);(3,12)(7,8)(13,16)(14,15)(17,19,21,26)(18,22,24,25)(20,27,30,28)(23,31,29,32)(33,35,39,37)(34,41,40,43)(36,47,46,44)(38,45,42,48);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,19)(18,22)(20,28)(21,26)(23,32)(24,25)(27,30)(29,31)(33,36,39,46)(34,42,40,38)(35,44,37,47)(41,45,43,48);(1,4)(2,9)(3,13)(5,11)(6,10)(7,15)(8,14)(12,16)(17,18)(19,22)(20,29)(21,24)(23,30)(25,26)(27,32)(28,31)(33,37,39,35)(34,43,40,41)(36,47,46,44)(38,45,42,48);(1,5,6,9)(2,10,11,4)(3,14,12,15)(7,16,8,13)(17,20,21,30)(18,23,24,29)(19,27,26,28)(22,31,25,32)(33,38,39,42)(34,36,40,46)(35,45,37,48)(41,47,43,44)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8653465);