#8621150

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

4020 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8528346);