#10422719

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	#49812	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10329915);