#9029119

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	#52477	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,d^2=h,e^2=g,f^2=i,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, 8936315);