#9164021

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	#58197	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4040 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:

(1,2,7,5)(3,8,11,13)(4,9,15,14)(6,10,16,12)(17,18,23,27)(19,28,26,22)(20,31,30,24)(21,25,29,32)(33,34,40,44)(35,39,37,42)(36,46,45,41)(38,43,47,48);(1,3,7,11)(2,8,5,13)(4,12,15,10)(6,14,16,9)(17,19,23,26)(18,22,27,28)(20,29,30,21)(24,32,31,25)(33,35,40,37)(34,39,44,42)(36,47,45,38)(41,43,46,48);(3,11)(4,15)(8,13)(9,14)(17,20,23,30)(18,24,27,31)(19,29,26,21)(22,32,28,25)(33,36,40,45)(34,41,44,46)(35,38,37,47)(39,48,42,43);(1,4)(2,9)(3,12)(5,14)(6,8)(7,15)(10,11)(13,16)(17,20)(18,24)(19,29)(21,26)(22,32)(23,30)(25,28)(27,31)(33,37,40,35)(34,42,44,39)(36,38,45,47)(41,48,46,43);(1,5,7,2)(3,13,11,8)(4,9,15,14)(6,10,16,12)(17,21,23,29)(18,25,27,32)(19,30,26,20)(22,31,28,24)(33,38)(34,43)(35,45)(36,37)(39,41)(40,47)(42,46)(44,48);(1,6)(2,10)(3,14)(4,8)(5,12)(7,16)(9,11)(13,15)(17,22)(18,26)(19,27)(20,32)(21,24)(23,28)(25,30)(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=gh,e^2=hi,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, 9071217);