#8551380

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

4085 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)(3,4)(5,10)(6,7)(8,9)(11,12)(13,16)(14,15)(17,18)(19,23)(20,25)(21,27)(22,24)(26,28)(29,31)(30,32)(33,34,39,38)(35,40,42,37)(36,41,45,46)(43,47,48,44);(2,7)(3,9)(10,15)(11,13)(17,19)(18,23)(20,27)(21,25)(22,28)(24,26)(29,30)(31,32)(33,35,39,42)(34,40,38,37)(36,43,45,48)(41,47,46,44);(2,7)(4,8)(10,15)(12,16)(18,24)(19,28)(25,32)(27,29)(35,42)(37,40)(43,48)(44,47);(1,3,6,9)(2,8,7,4)(5,13,14,11)(10,12,15,16)(17,20)(18,25)(19,29)(21,26)(22,30)(23,31)(24,32)(27,28)(33,36)(34,41)(35,43)(37,44)(38,46)(39,45)(40,47)(42,48);(1,4,6,8)(2,9,7,3)(5,12,14,16)(10,13,15,11)(17,19)(18,26)(20,29)(21,25)(22,28)(23,24)(27,30)(31,32)(33,37)(34,35)(36,44)(38,42)(39,40)(41,43)(45,47)(46,48);(1,5)(2,10)(3,11)(4,12)(6,14)(7,15)(8,16)(9,13)(17,21)(18,27)(19,25)(20,23)(22,31)(24,29)(26,30)(28,32)(33,38,39,34)(35,37,42,40)(36,41,45,46)(43,47,48,44)

Polycyclic presentation:

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

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, 8458576);