#8739301

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

3818 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=h,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, 8646497);