#8463504

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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)(5,9,14,13)(11,16,12,15)(17,18,23,27)(19,28,29,24)(20,25,31,30)(21,22,26,32)(33,34,39,44)(35,45,47,40)(36,37,42,41)(38,43,46,48);(1,3,6,4)(2,7,10,8)(5,11,14,12)(9,15,13,16)(19,29)(20,31)(24,28)(25,30)(33,35)(34,40)(36,46)(37,43)(38,42)(39,47)(41,48)(44,45);(17,19,23,29)(18,24,27,28)(20,26,31,21)(22,30,32,25)(33,36,39,42)(34,41,44,37)(35,46,47,38)(40,48,45,43);(1,2)(3,7)(4,8)(5,13)(6,10)(9,14)(11,16)(12,15)(17,20)(18,25)(19,26)(21,29)(22,24)(23,31)(27,30)(28,32)(33,34)(35,45)(36,41)(37,42)(38,48)(39,44)(40,47)(43,46);(1,4,6,3)(2,8,10,7)(5,12,14,11)(9,16,13,15)(17,21,23,26)(18,22,27,32)(19,20,29,31)(24,30,28,25)(33,37,39,41)(34,42,44,36)(35,43,47,48)(38,45,46,40);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,16)(10,13)(17,22)(18,26)(19,30)(20,28)(21,27)(23,32)(24,31)(25,29)(33,38,39,46)(34,43,44,48)(35,42,47,36)(37,45,41,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8370700);