#8521617

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

3950 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=g,e^2=g,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, 8428813);