#9717421

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

4313 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=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, 9624617);