#9034485

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

4166 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)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18,23,21)(19,24,20,25)(22,26,30,29)(27,31,28,32)(33,34)(35,38)(36,44)(37,41)(39,43)(40,45)(42,47)(46,48);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(18,21)(24,25)(26,29)(31,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(3,11)(5,15)(6,16)(9,14)(17,19,23,20)(18,24,21,25)(22,28,30,27)(26,32,29,31)(33,34,37,41)(35,39,43,38)(36,44,45,40)(42,46,48,47);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,21,23,18)(19,24,20,25)(22,29,30,26)(27,31,28,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48)

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8941681);