#8934223

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	#50354	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4103 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)(4,8)(5,6)(7,11)(10,16)(12,15)(13,14)(17,18,22,26)(19,23,27,31)(20,29,30,24)(21,28,32,25)(33,34,39,41)(35,40,44,47)(36,37,46,42)(38,45,48,43);(1,3)(2,7)(4,12)(5,9)(6,11)(8,16)(10,13)(14,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,35,39,44)(34,40,41,47)(36,43,46,45)(37,38,42,48);(2,5)(4,13)(7,9)(10,12)(19,27)(20,30)(23,31)(24,29)(34,41)(35,44)(36,46)(38,48);(1,4,6,13)(2,8,5,14)(3,10,11,12)(7,15,9,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,36,39,46)(34,42,41,37)(35,45,44,43)(38,47,48,40);(1,3)(2,7)(4,12)(5,9)(6,11)(8,16)(10,13)(14,15)(17,21)(18,25)(19,24)(20,31)(22,32)(23,30)(26,28)(27,29)(33,37,39,42)(34,36,41,46)(35,38,44,48)(40,43,47,45);(1,5,6,2)(3,9,11,7)(4,14,13,8)(10,16,12,15)(17,19)(18,23)(20,28)(21,24)(22,27)(25,30)(26,31)(29,32)(33,38)(34,43)(35,42)(36,47)(37,44)(39,48)(40,46)(41,45)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=h,e^2=gh,f^2=g,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, 8841419);