#9054503

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

4121 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)(3,4)(5,8)(6,7)(9,11)(10,12)(13,14)(15,16)(17,18,23,21)(19,24,20,26)(22,29,30,25)(27,32,28,31)(33,34,39,43)(35,40,44,38)(36,37,42,41)(45,47,48,46);(2,7)(4,11)(8,14)(12,16)(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32)(33,35,39,44)(34,40,43,38)(36,45,42,48)(37,47,41,46);(3,9)(4,11)(5,13)(8,14)(18,21)(19,20)(22,30)(31,32)(35,44)(36,42)(37,41)(38,40);(1,3,6,9)(2,4,7,11)(5,10,13,15)(8,12,14,16)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,36)(34,41)(35,45)(37,43)(38,47)(39,42)(40,46)(44,48);(1,4)(2,3)(5,12)(6,11)(7,9)(8,10)(13,16)(14,15)(17,21,23,18)(19,26,20,24)(22,29,30,25)(27,32,28,31)(33,37,39,41)(34,42,43,36)(35,46,44,47)(38,48,40,45);(1,5,6,13)(2,8,7,14)(3,10,9,15)(4,12,11,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38)(34,35)(36,47)(37,48)(39,40)(41,45)(42,46)(43,44)

Polycyclic presentation:

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

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, 8961699);