#9040553

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

4176 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947749);