#8934318

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	#51930	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) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=h,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, 8841514);