#8201989

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

4349 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)(5,9)(6,7)(8,11)(10,12)(13,14)(15,16)(17,18,24,28)(19,29,31,25)(20,21,22,26)(23,27,32,30)(33,34)(35,36)(37,45)(38,40)(39,47)(41,43)(42,48)(44,46);(2,7)(4,8)(9,14)(12,15)(17,19,24,31)(18,25,28,29)(20,30,22,27)(21,32,26,23)(33,35,38,43)(34,36,40,41)(37,46,47,42)(39,48,45,44);(5,13)(9,14)(10,16)(12,15)(17,20,24,22)(18,26,28,21)(19,30,31,27)(23,29,32,25)(33,34,38,40)(35,41,43,36)(37,39,47,45)(42,48,46,44);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,21)(18,20)(19,23)(22,28)(24,26)(25,27)(29,30)(31,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,46)(47,48);(1,4,6,8)(2,3,7,11)(5,12,13,15)(9,10,14,16)(17,22,24,20)(18,21,28,26)(19,30,31,27)(23,29,32,25)(33,37)(34,39)(35,42)(36,44)(38,47)(40,45)(41,48)(43,46);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,23)(18,27)(19,21)(20,25)(22,29)(24,32)(26,31)(28,30)(33,34)(35,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^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, 8109185);