#10181046

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

4059 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10088242);