#10168358

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

4072 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=gi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=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, 10075554);