#9913153

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

4024 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)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,18,23,21)(19,24,20,25)(22,26,30,29)(27,31,28,32)(35,44)(37,47)(39,40)(42,45);(3,9)(6,13)(10,15)(14,16)(18,21)(24,25)(26,29)(31,32)(33,34,38,43)(35,40,44,39)(36,41,46,48)(37,42,47,45);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35,38,44)(34,39,43,40)(36,45,46,42)(37,41,47,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,35)(34,40)(36,45)(37,41)(38,44)(39,43)(42,46)(47,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,21,23,18)(19,25,20,24)(22,26,30,29)(27,31,28,32)(33,36)(34,41)(35,45)(37,40)(38,46)(39,47)(42,44)(43,48);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,15)(13,16)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,37)(34,42)(35,41)(36,39)(38,47)(40,46)(43,45)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=g,e^2=g,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, 9820349);