#10506329

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

4051 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=g,e^2=hi,f^2=h,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, 10413525);