#8202442

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

4227 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=h,e^2=g,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, 8109638);