#8270518

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

4208 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8177714);