#10320337

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

3833 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)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(18,24)(19,20)(26,32)(27,28)(33,34,39,37)(35,40,43,36)(38,45,47,41)(42,44,46,48);(4,11)(8,14)(10,15)(12,16)(17,18)(19,25)(20,21)(22,26)(23,24)(27,30)(28,29)(31,32)(33,35,39,43)(34,40,37,36)(38,42,47,46)(41,48,45,44);(2,7)(3,9)(8,14)(10,15)(17,19,23,20)(18,25,24,21)(22,27,31,28)(26,30,32,29)(33,36,39,40)(34,35,37,43)(38,44,47,48)(41,42,45,46);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,30,32,29)(33,37,39,34)(35,40,43,36)(38,45,47,41)(42,48,46,44);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,21,23,25)(18,20,24,19)(22,30,31,29)(26,27,32,28)(33,35)(34,40)(36,37)(38,46)(39,43)(41,44)(42,47)(45,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22,23,31)(18,26,24,32)(19,27,20,28)(21,29,25,30)(33,38)(34,41)(35,42)(36,44)(37,45)(39,47)(40,48)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=hi,e^2=h,f^2=g,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, 10227533);