#8739303

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3817 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)(3,9,14,7)(4,5,15,8)(11,16,12,13)(18,20)(23,26)(24,28)(30,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,48)(45,47);(1,3)(2,7)(4,11)(5,13)(6,14)(8,16)(9,10)(12,15)(19,25)(23,26)(27,31)(30,32)(33,34,38,42)(35,39,46,43)(36,40,37,41)(44,47,45,48);(1,4,6,15)(2,8,10,5)(3,11,14,12)(7,16,9,13)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4)(2,5)(3,12)(6,15)(7,16)(8,10)(9,13)(11,14)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,35,38,46)(34,39,42,43)(36,44,37,45)(40,47,41,48);(1,5)(2,4)(3,13)(6,8)(7,11)(9,12)(10,15)(14,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,36,38,37)(34,40,42,41)(35,44,46,45)(39,47,43,48);(1,3)(2,9)(4,11)(5,16)(6,14)(7,10)(8,13)(12,15)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,37,38,36)(34,41,42,40)(35,45,46,44)(39,48,43,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8646499);