#8192514

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

4130 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(33,34,38,39)(35,37,41,46)(36,43,45,40)(42,48,47,44);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,18,22,23)(19,25,27,21)(20,24,29,30)(26,31,32,28)(33,34,38,39)(35,37,41,46)(36,40,45,43)(42,44,47,48);(6,14)(9,12)(11,13)(15,16)(18,23)(21,25)(24,30)(28,31)(34,39)(37,46)(40,43)(44,48);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,35,38,41)(34,37,39,46)(36,42,45,47)(40,44,43,48);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,20,22,29)(18,24,23,30)(19,26,27,32)(21,28,25,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,43)(41,47)(46,48);(1,6,7,14)(2,9,4,12)(3,11,5,13)(8,15,10,16)(17,21)(18,19)(20,28)(22,25)(23,27)(24,26)(29,31)(30,32)(33,37)(34,41)(35,39)(36,44)(38,46)(40,47)(42,43)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8099710);