#9224355

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

4164 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)(3,8)(4,14)(5,6)(7,10)(9,13)(11,16)(12,15)(17,18)(19,28)(20,25)(21,30)(22,26)(23,27)(24,31)(29,32)(33,34,39,42)(35,36,37,40)(38,46,48,41)(43,47,45,44);(1,3)(2,8)(4,15)(5,11)(6,16)(7,13)(9,10)(12,14)(17,19,23,31)(18,24,27,28)(20,29,21,26)(22,30,32,25)(33,35,39,37)(34,40,42,36)(38,43,48,45)(41,47,46,44);(1,4,7,5)(2,6,10,14)(3,11,13,15)(8,12,9,16)(17,20,23,21)(18,25,27,30)(19,29,31,26)(22,28,32,24)(33,36)(34,35)(37,42)(38,44)(39,40)(41,43)(45,46)(47,48);(1,5,7,4)(2,6,10,14)(3,11,13,15)(8,16,9,12)(17,21,23,20)(18,25,27,30)(19,26,31,29)(22,28,32,24)(33,37,39,35)(34,36,42,40)(38,43,48,45)(41,47,46,44);(1,6)(2,5)(3,12)(4,10)(7,14)(8,15)(9,11)(13,16)(17,18)(19,24)(20,25)(21,30)(22,29)(23,27)(26,32)(28,31)(33,36)(34,37)(35,42)(38,47)(39,40)(41,43)(44,48)(45,46);(1,3)(2,9)(4,11)(5,15)(6,16)(7,13)(8,10)(12,14)(17,22,23,32)(18,26,27,29)(19,30,31,25)(20,28,21,24)(33,38,39,48)(34,41,42,46)(35,43,37,45)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=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, 9131551);