#8115294

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	#34633	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4514 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)(9,10,13,11)(12,15,16,14)(17,18)(19,27)(20,30)(21,33)(22,36)(23,24)(25,34)(26,37)(28,31)(29,32)(35,38)(39,41)(40,43)(42,47)(44,48)(45,46);(3,7)(6,8)(10,11)(14,15)(18,24)(20,27)(21,34)(26,36)(28,41)(32,40)(35,46)(42,48);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19,23,30)(18,20,24,27)(21,28,34,41)(22,29,37,43)(25,31,33,39)(26,32,36,40)(35,42,46,48)(38,44,45,47);(1,3)(2,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,19)(21,31)(22,32)(23,27)(24,30)(25,28)(26,29)(33,41)(34,39)(35,44)(36,43)(37,40)(38,42)(45,48)(46,47);(1,2)(3,6)(4,5)(7,8)(17,21)(18,25)(19,28)(20,31)(22,35)(23,34)(24,33)(26,38)(27,39)(29,42)(30,41)(32,44)(36,45)(37,46)(40,47)(43,48);(17,22,23,37)(18,26,24,36)(19,29,30,43)(20,32,27,40)(21,35,34,46)(25,38,33,45)(28,42,41,48)(31,44,39,47)

Polycyclic presentation:

2^9|a^2=i,c^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, 8022490);