#10247258

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

4002 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154454);