#10320374

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=gi,e^2=g,f^2=h,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, 10227570);