#9159929

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	#51967	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4111 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)(3,8)(4,12)(5,9)(6,11)(7,13)(10,15)(14,16)(17,18)(19,23)(20,25)(21,28)(22,24)(26,32)(27,30)(29,31)(33,34,37,40)(35,41,43,38)(36,39,46,44)(42,48,45,47);(1,3)(2,6)(4,10)(5,11)(7,15)(8,9)(12,16)(13,14)(17,19,22,27)(18,23,24,30)(20,26,31,28)(21,25,32,29)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(4,13)(7,12)(10,14)(15,16)(18,24)(19,27)(25,29)(26,28)(33,35,37,43)(34,38,40,41)(36,42,46,45)(39,47,44,48);(1,4,5,13)(2,7,9,12)(3,10,11,14)(6,15,8,16)(17,18)(19,23)(20,29)(21,26)(22,24)(25,31)(27,30)(28,32)(33,35)(34,38)(36,45)(37,43)(39,48)(40,41)(42,46)(44,47);(1,3)(2,8)(4,14)(5,11)(6,9)(7,15)(10,13)(12,16)(17,20)(18,25)(19,28)(21,23)(22,31)(24,29)(26,27)(30,32)(33,36)(34,39)(35,42)(37,46)(38,47)(40,44)(41,48)(43,45);(1,2)(3,6)(4,7)(5,9)(8,11)(10,15)(12,13)(14,16)(17,21)(18,26)(19,25)(20,30)(22,32)(23,31)(24,28)(27,29)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=ghi,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, 9067125);