#8115296

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	#37656	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4537 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022492);