#10320381

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

3860 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)(3,8,13,7)(4,14,15,5)(10,12,11,16)(17,18)(19,28)(20,24)(21,32)(22,27)(23,30)(25,31)(26,29)(33,34)(35,40)(36,42)(37,41)(38,43)(39,44)(45,46)(47,48);(1,3)(2,7)(4,10)(5,12)(6,13)(8,9)(11,15)(14,16)(17,19)(18,23)(20,21)(22,30)(24,26)(25,32)(27,28)(29,31)(33,35,39,47)(34,40,44,48)(36,37,43,45)(38,46,42,41);(1,4)(2,5)(3,10)(6,15)(7,12)(8,16)(9,14)(11,13)(17,20,22,31)(18,24,27,25)(19,21,30,29)(23,26,28,32)(33,36)(34,38)(35,45)(37,47)(39,43)(40,41)(42,44)(46,48);(1,4)(2,5)(3,11)(6,15)(7,16)(8,12)(9,14)(10,13)(17,20)(18,25)(19,29)(21,30)(22,31)(23,26)(24,27)(28,32)(33,37,39,45)(34,41,44,46)(35,36,47,43)(38,40,42,48);(1,3)(2,8)(4,10)(5,16)(6,13)(7,9)(11,15)(12,14)(17,18)(19,28)(20,24)(21,32)(22,27)(23,30)(25,31)(26,29)(33,36)(34,42)(35,37)(38,44)(39,43)(40,41)(45,47)(46,48);(1,5)(2,4)(3,12)(6,14)(7,10)(8,11)(9,15)(13,16)(17,21,22,29)(18,26,27,32)(19,20,30,31)(23,24,28,25)(33,38,39,42)(34,43,44,36)(35,46,47,41)(37,40,45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,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, 10227577);