#10521335

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

4116 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=ghi,e^2=gi,f^2=g,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, 10428531);