#9913606

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820802);