#8934858

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	#47807	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,18,23,26)(19,24,27,32)(20,30,29,22)(21,31,28,25)(33,34,40,43)(35,41,39,48)(36,37,38,42)(44,45,46,47);(2,6)(7,9)(8,12)(14,15)(17,19)(18,24)(20,21)(22,25)(23,27)(26,32)(28,29)(30,31)(33,35,40,39)(34,41,43,48)(36,46,38,44)(37,47,42,45);(2,6)(4,13)(7,9)(10,16)(19,27)(20,29)(22,30)(24,32)(33,36,40,38)(34,42,43,37)(35,44,39,46)(41,47,48,45);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20)(18,22)(19,28)(21,27)(23,29)(24,31)(25,32)(26,30)(33,37,40,42)(34,36,43,38)(35,45,39,47)(41,44,48,46);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21,23,28)(18,25,26,31)(19,29,27,20)(22,24,30,32)(33,38,40,36)(34,37,43,42)(35,44,39,46)(41,47,48,45);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,22)(18,20)(19,25)(21,24)(23,30)(26,29)(27,31)(28,32)(33,39,40,35)(34,41,43,48)(36,44,38,46)(37,47,42,45)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8842054);