#8828651

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

4293 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8735847);