#8617755

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

4005 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=gh,e^2=hi,f^2=gh,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, 8524951);