#9059300

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

4161 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8966496);