#8114373

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3983 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gi,e^2=i,f^2=g,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, 8021569);