#8934315

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	#51392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=gh,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, 8841511);