#8188514

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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)(3,5)(4,12)(6,8)(7,11)(9,14)(10,15)(13,16)(18,21)(19,27)(20,30)(26,32)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(1,3)(2,7)(4,10)(5,8)(6,11)(9,15)(12,13)(14,16)(17,18,23,21)(19,28,27,24)(20,31,30,25)(22,26,29,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(2,8)(5,7)(9,12)(13,15)(17,19)(18,24)(20,29)(21,28)(22,30)(23,27)(25,32)(26,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,3)(2,7)(4,10)(5,8)(6,11)(9,15)(12,13)(14,16)(17,20,23,30)(18,25,21,31)(19,29,27,22)(24,32,28,26);(1,4,6,14)(2,9,8,12)(3,10,11,16)(5,13,7,15)(17,21,23,18)(19,28,27,24)(20,31,30,25)(22,32,29,26)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47);(1,5,6,7)(2,3,8,11)(4,13,14,15)(9,10,12,16)(17,22,23,29)(18,26,21,32)(19,30,27,20)(24,31,28,25)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=ghi,e^2=h,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, 8095710);