#8934320

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	#49887	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,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, 8841516);