#8934892

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	#2818	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(19,23)(20,29)(21,26)(22,27)(24,28)(25,31)(30,32)(35,37)(36,44)(38,40)(41,45);(2,6)(7,9)(8,12)(14,15)(17,19,22,28)(18,23,27,24)(20,30,31,26)(21,29,32,25)(33,34)(35,38)(36,45)(37,40)(39,42)(41,44)(43,48)(46,47);(2,6)(4,13)(7,9)(10,16)(19,28)(21,32)(23,24)(26,30)(33,35,39,37)(34,40,42,38)(36,46,44,43)(41,47,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19)(18,24)(20,30)(21,29)(22,28)(23,27)(25,32)(26,31)(33,36,39,44)(34,41,42,45)(35,43,37,46)(38,47,40,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20,22,31)(18,25,27,29)(19,26,28,30)(21,23,32,24)(33,37,39,35)(34,40,42,38)(36,43,44,46)(41,47,45,48);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,21)(18,26)(19,29)(20,23)(22,32)(24,31)(25,28)(27,30)(33,38,39,40)(34,35,42,37)(36,47,44,48)(41,43,45,46)

Polycyclic presentation:

2^9|b^2=ghi,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, 8842088);