#9015641

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	#40953	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3917 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)(4,10)(5,11)(6,16)(7,9)(12,15)(13,14)(17,18,23,21)(19,24,29,27)(20,25,31,22)(26,28,30,32)(33,34,38,42)(35,41,45,39)(36,46,40,37)(43,48,47,44);(1,3,7,6)(2,8,9,16)(4,12,14,11)(5,10,15,13)(17,19)(18,24)(20,30)(21,27)(22,28)(23,29)(25,32)(26,31)(33,35,38,45)(34,39,42,41)(36,47,40,43)(37,44,46,48);(2,9)(3,6)(4,14)(5,15)(17,20,23,31)(18,25,21,22)(19,26,29,30)(24,28,27,32)(33,36,38,40)(34,37,42,46)(35,43,45,47)(39,48,41,44);(1,4)(2,10)(3,12)(5,16)(6,11)(7,14)(8,15)(9,13)(17,21,23,18)(19,27,29,24)(20,25,31,22)(26,28,30,32)(33,37)(34,40)(35,44)(36,42)(38,46)(39,43)(41,47)(45,48);(1,5,7,15)(2,11,9,12)(3,13,6,10)(4,8,14,16)(17,22,23,25)(18,20,21,31)(19,28,29,32)(24,30,27,26)(33,35)(34,41)(36,43)(37,44)(38,45)(39,42)(40,47)(46,48);(1,6,7,3)(2,8,9,16)(4,11,14,12)(5,10,15,13)(17,19)(18,24)(20,30)(21,27)(22,28)(23,29)(25,32)(26,31)(33,37)(34,40)(35,44)(36,42)(38,46)(39,43)(41,47)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=hi,e^2=gi,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, 8922837);