#8934193

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	#35422	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4134 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=gh,e^2=h,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, 8841389);