#8934854

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	#2615	C₂²
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:

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

Polycyclic presentation:

2^9|a^2=hi,b^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, 8842050);