#8093466

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

4159 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)(3,4)(5,13)(6,7)(8,14)(9,11)(10,16)(12,15)(17,18)(19,26)(20,21)(22,23)(24,27)(25,30)(28,32)(29,31)(36,44)(40,45)(42,48)(46,47);(3,9)(4,11)(10,15)(12,16)(18,23)(20,25)(24,26)(28,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,7)(3,9)(8,13)(10,15)(19,27)(24,26)(28,31)(29,32)(34,38)(39,41)(40,45)(46,47);(1,3)(2,4)(5,10)(6,9)(7,11)(8,12)(13,16)(14,15)(17,19)(18,24)(20,28)(21,29)(22,27)(23,26)(25,31)(30,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4)(2,3)(5,12)(6,11)(7,9)(8,10)(13,15)(14,16)(17,20,22,25)(18,21,23,30)(19,28,27,31)(24,29,26,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,6,14)(2,8,7,13)(3,10,9,15)(4,12,11,16)(17,21)(18,25)(19,29)(20,23)(22,30)(24,31)(26,28)(27,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

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, 8000662);