#8839568

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

4160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=ghi,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, 8746764);