#8202383

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

4333 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=gh,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, 8109579);