#9127691

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	#49754	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4220 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=g,e^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, 9034887);