#8095480

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	#52451	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=ghi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002676);