#9164024

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	#51086	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4186 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9071220);