#8841668

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	#53410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8748864);