#8202455

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

4282 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,23,24)(19,21,20,25)(22,26,30,31)(27,29,28,32)(34,40)(35,36)(37,48)(44,47);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(18,24)(19,20)(26,31)(27,28)(33,34)(35,41)(36,45)(37,38)(39,40)(42,48)(43,47)(44,46);(4,12)(7,14)(10,16)(13,15)(17,19,23,20)(18,25,24,21)(22,27,30,28)(26,32,31,29)(33,35,39,36)(34,41,40,45)(37,43,48,46)(38,47,42,44);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20,23,19)(18,21,24,25)(22,28,30,27)(26,29,31,32)(33,36,39,35)(34,41,40,45)(37,46,48,43)(38,47,42,44);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,21)(18,19)(20,24)(22,29)(23,25)(26,27)(28,31)(30,32)(33,37)(34,42)(35,43)(36,46)(38,40)(39,48)(41,44)(45,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,22,23,30)(18,26,24,31)(19,27,20,28)(21,29,25,32)(33,38,39,42)(34,37,40,48)(35,44,36,47)(41,46,45,43)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=h,f^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, 8109651);