#8633543

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

4088 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8540739);