#8695990

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

4199 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)(3,11)(4,9)(5,10)(6,7)(8,12)(13,14)(15,16)(17,18,21,24)(19,25,27,22)(20,23,30,28)(26,32,29,31)(35,37)(39,42)(43,44)(47,48);(2,7)(3,12)(9,14)(10,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,31)(28,30)(29,32)(33,34)(35,42)(36,40)(37,39)(38,41)(43,48)(44,47)(45,46);(17,18)(19,22)(20,23)(21,24)(25,27)(26,31)(28,30)(29,32)(33,35,38,37)(34,39,41,42)(36,43,46,44)(40,47,45,48);(1,3)(2,8)(4,10)(5,14)(6,12)(7,11)(9,15)(13,16)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,36)(34,40)(35,43)(37,44)(38,46)(39,47)(41,45)(42,48);(1,4)(2,9)(3,10)(5,11)(6,13)(7,14)(8,15)(12,16)(17,19,21,27)(18,22,24,25)(20,29,30,26)(23,32,28,31)(33,37,38,35)(34,39,41,42)(36,44,46,43)(40,47,45,48);(1,5,6,15)(2,10,7,16)(3,9,12,14)(4,8,13,11)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32)(33,34)(35,39)(36,45)(37,42)(38,41)(40,46)(43,48)(44,47)

Polycyclic presentation:

2^9|a^2=hi,c^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603186);