#10195848

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 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)(3,11)(4,13)(5,10)(6,14)(7,9)(8,12)(15,16)(17,18)(19,26)(20,25)(21,22)(23,27)(24,31)(28,30)(29,32)(33,34,39,44)(35,43,38,40)(36,46,47,41)(37,42,48,45);(1,3,7,12)(2,8,9,11)(4,5,14,15)(6,10,13,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,27)(33,35,39,38)(34,40,44,43)(36,45,47,42)(37,46,48,41);(2,9)(3,12)(4,14)(10,16)(17,20,23,21)(18,25,27,22)(19,28,31,29)(24,32,26,30)(33,36,39,47)(34,41,44,46)(35,45,38,42)(37,43,48,40);(1,4)(2,6)(3,5)(7,14)(8,10)(9,13)(11,16)(12,15)(17,21,23,20)(18,25,27,22)(19,29,31,28)(24,32,26,30)(33,36)(34,41)(35,45)(37,40)(38,42)(39,47)(43,48)(44,46);(1,5)(2,10)(3,4)(6,8)(7,15)(9,16)(11,13)(12,14)(17,22,23,25)(18,21,27,20)(19,30,31,32)(24,29,26,28)(33,37)(34,42)(35,41)(36,43)(38,46)(39,48)(40,47)(44,45);(1,6)(2,4)(3,10)(5,8)(7,13)(9,14)(11,15)(12,16)(17,19)(18,26)(20,28)(21,29)(22,32)(23,31)(24,27)(25,30)(33,38,39,35)(34,43,44,40)(36,42,47,45)(37,41,48,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=gh,e^2=gh,f^2=hi,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, 10103044);