#8534553

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

4367 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8441749);