#10320392

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	#42620	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3981 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,22,23)(19,26,20,24)(21,25,29,30)(27,32,28,31)(33,34,39,37)(35,42,41,38)(36,44,47,40)(43,46,45,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(18,23)(24,26)(25,30)(31,32)(33,35,39,41)(34,38,37,42)(36,45,47,43)(40,48,44,46);(2,3)(7,9)(8,10)(14,15)(17,19,22,20)(18,24,23,26)(21,27,29,28)(25,31,30,32)(33,34,39,37)(35,38,41,42)(36,44,47,40)(43,48,45,46);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,18)(19,24)(20,26)(21,25)(22,23)(27,31)(28,32)(29,30)(33,36,39,47)(34,40,37,44)(35,43,41,45)(38,46,42,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,22,19)(18,24,23,26)(21,27,29,28)(25,32,30,31)(33,37,39,34)(35,38,41,42)(36,40,47,44)(43,48,45,46);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,29)(18,25,23,30)(19,27,20,28)(24,31,26,32)(33,38)(34,41)(35,37)(36,46)(39,42)(40,45)(43,44)(47,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gi,d^2=ghi,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, 10227588);