#9062096

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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)(4,15)(5,10)(6,7)(8,14)(9,13)(12,16)(17,18,23,26)(19,24,27,22)(20,21,25,32)(28,29,30,31)(33,34,38,42)(35,40,45,39)(36,37,41,48)(43,44,46,47);(1,3,7,14)(2,8,6,11)(4,5,13,12)(9,10,15,16)(17,19,23,27)(18,24,26,22)(20,30,25,28)(21,31,32,29)(33,35,38,45)(34,39,42,40)(36,46,41,43)(37,44,48,47);(2,6)(4,13)(5,12)(8,11)(19,27)(20,25)(21,32)(22,24)(33,35,38,45)(34,40,42,39)(36,46,41,43)(37,47,48,44);(1,4)(2,9)(3,12)(5,14)(6,15)(7,13)(8,16)(10,11)(17,20)(18,21)(19,28)(22,31)(23,25)(24,29)(26,32)(27,30)(33,35)(34,39)(36,46)(37,44)(38,45)(40,42)(41,43)(47,48);(1,5,7,12)(2,10,6,16)(3,13,14,4)(8,15,11,9)(17,21)(18,25)(19,29)(20,26)(22,28)(23,32)(24,30)(27,31)(33,36,38,41)(34,37,42,48)(35,43,45,46)(39,47,40,44);(1,6,7,2)(3,11,14,8)(4,15,13,9)(5,16,12,10)(17,22)(18,19)(20,31)(21,28)(23,24)(25,29)(26,27)(30,32)(33,37)(34,41)(35,44)(36,42)(38,48)(39,43)(40,46)(45,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969292);