#8749868

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	#40053	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4191 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=g,f^2=gi,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, 8657064);