#8534653

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

4091 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,d^2=h,e^2=i,f^2=g,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, 8441849);