#10534329

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

4298 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gi,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10441525);