#8893652

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

4160 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8800848);