#9017694

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	#48143	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4251 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)(3,11)(4,9)(5,15)(6,7)(8,12)(10,16)(13,14)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32)(34,36)(39,43)(40,44)(47,48);(2,7)(4,14)(8,11)(10,15)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32)(33,34,38,36)(35,39,42,43)(37,40,46,44)(41,47,45,48);(2,7)(3,12)(4,14)(5,16)(18,22)(19,26)(20,29)(30,31)(33,35,38,42)(34,39,36,43)(37,45,46,41)(40,48,44,47);(1,3)(2,8)(4,15)(5,13)(6,12)(7,11)(9,16)(10,14)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32)(33,36,38,34)(35,39,42,43)(37,44,46,40)(41,47,45,48);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,11,16,8)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,47)(44,48)(45,46);(1,5)(2,10)(3,13)(4,11)(6,16)(7,15)(8,14)(9,12)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,37,38,46)(34,40,36,44)(35,41,42,45)(39,47,43,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=ghi,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^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924890);