#8549578

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

4122 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,e^2=gh,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, 8456774);