#8197342

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	#48681	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4102 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104538);