#8417292

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	#2818	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3845 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=hi,e^2=gi,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, 8324488);