#8934332

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	#49797	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4118 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)(19,22)(20,21)(23,25)(24,27)(26,31)(28,29)(30,32)(35,37)(36,44)(38,42)(40,45);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,19,23,27)(18,24,25,22)(20,30,26,28)(21,29,31,32)(33,34,39,41)(35,42,37,38)(36,45,44,40)(43,47,46,48);(3,11)(4,14)(7,15)(8,13)(18,25)(20,26)(22,24)(28,30)(33,35,39,37)(34,38,41,42)(36,46,44,43)(40,48,45,47);(1,4)(2,8)(3,12)(5,7)(6,14)(9,11)(10,13)(15,16)(17,20)(18,21)(19,28)(22,29)(23,26)(24,32)(25,31)(27,30)(33,36,39,44)(34,40,41,45)(35,43,37,46)(38,47,42,48);(1,5)(2,9)(3,8)(4,15)(6,16)(7,14)(10,12)(11,13)(17,21)(18,26)(19,29)(20,25)(22,30)(23,31)(24,28)(27,32)(33,37,39,35)(34,38,41,42)(36,43,44,46)(40,48,45,47);(1,3)(2,7)(4,12)(5,8)(6,11)(9,14)(10,15)(13,16)(17,22,23,24)(18,27,25,19)(20,29,26,32)(21,30,31,28)(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=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, 8841528);