#8417328

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	#52757	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324524);