#8253900

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

4157 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)(3,7)(4,12)(5,6)(8,13)(9,11)(10,15)(14,16)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(34,40)(35,36)(37,47)(43,46);(1,2)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(33,34,39,40)(35,41,36,44)(37,48,47,38)(42,43,45,46);(2,6)(7,9)(8,12)(14,15)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,35,39,36)(34,41,40,44)(37,42,47,45)(38,46,48,43);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,36,39,35)(34,41,40,44)(37,45,47,42)(38,46,48,43);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31)(33,37)(34,38)(35,42)(36,45)(39,47)(40,48)(41,46)(43,44);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,38)(34,37)(35,43)(36,46)(39,48)(40,47)(41,45)(42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8161096);