#8622317

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

4166 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(18,19)(23,25)(24,26)(30,31)(33,34,39,41)(35,40,37,38)(36,42,46,48)(43,45,44,47);(2,7)(3,11)(4,14)(13,15)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30)(33,35)(34,40)(36,44)(37,39)(38,41)(42,47)(43,46)(45,48);(2,7)(4,14)(8,10)(12,16)(20,27)(23,25)(28,32)(30,31)(34,41)(35,37)(42,48)(43,44);(1,3)(2,8)(4,12)(5,15)(6,11)(7,10)(9,13)(14,16)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,36,39,46)(34,42,41,48)(35,43,37,44)(38,47,40,45);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,37,39,35)(34,40,41,38)(36,44,46,43)(42,45,48,47);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,40)(34,35,41,37)(36,45,46,47)(42,44,48,43)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=gh,e^2=i,f^2=h,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, 8529513);