#8393698

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32);(2,3)(4,11)(8,10)(12,16)(17,19)(18,23)(20,29)(21,25)(22,27)(24,32)(26,30)(28,31)(33,34,38,39)(35,41,44,40)(36,45,47,37)(42,48,46,43);(5,13)(8,10)(12,16)(14,15)(18,21)(23,25)(24,28)(31,32)(34,39)(37,45)(40,41)(43,48);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,23)(20,29)(21,25)(22,27)(24,32)(26,30)(28,31)(33,35,38,44)(34,40,39,41)(36,46,47,42)(37,43,45,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,32)(27,29)(33,36)(34,37)(35,42)(38,47)(39,45)(40,48)(41,43)(44,46);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(33,37)(34,36)(35,43)(38,45)(39,47)(40,46)(41,42)(44,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8300894);