#8458080

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

4206 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8365276);