#9481434

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

4407 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388630);