#8115504

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	#42652	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4467 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:

(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,31)(21,23)(22,35)(24,30)(25,32)(26,36)(28,37)(29,38)(33,39)(34,40)(41,43)(42,48)(44,45)(46,47);(2,5)(6,7)(9,10,12,14)(11,13,16,15)(18,21)(20,32)(24,27)(26,35)(28,33)(34,45)(38,41)(42,46);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,24)(20,33)(21,27)(22,29)(23,30)(25,39)(26,38)(28,32)(31,37)(34,46)(35,41)(36,43)(40,48)(42,45)(44,47);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,28)(21,31)(22,34)(23,32)(24,37)(26,40)(27,39)(29,42)(30,33)(35,44)(36,45)(38,47)(41,48)(43,46);(1,3,4,8)(2,6,5,7)(17,21,23,18)(19,27,30,24)(20,31,32,25)(22,35,36,26)(28,39,33,37)(29,41,43,38)(34,44,45,40)(42,48,46,47);(17,22,23,36)(18,26,21,35)(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=ghi,b^2=gh,d^2=gh,e^2=h,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, 8022700);