#9839332

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
prime radical	#54847	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4108 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,20)(19,21,24,26)(22,28,31,25)(27,30,32,29)(34,40)(35,38)(36,42)(45,48);(4,11)(8,14)(10,15)(12,16)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32)(33,34,39,40)(35,41,38,43)(36,37,42,47)(44,48,46,45);(2,7)(3,9)(8,14)(10,15)(18,20)(19,24)(25,28)(27,32)(33,35,39,38)(34,41,40,43)(36,46,42,44)(37,45,47,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,18)(19,26,24,21)(22,28,31,25)(27,30,32,29)(33,36,39,42)(34,37,40,47)(35,44,38,46)(41,48,43,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,23,26)(18,24,20,19)(22,30,31,29)(25,27,28,32)(33,37)(34,42)(35,45)(36,40)(38,48)(39,47)(41,44)(43,46);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,38,39,35)(34,43,40,41)(36,46,42,44)(37,45,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9746528);