#8893433

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
prime radical	#38773	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4102 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)(17,18,22,26)(19,27,28,23)(20,24,29,32)(21,25,30,31)(33,34,39,44)(35,43,46,40)(36,47,48,41)(37,38,45,42);(1,3)(2,7)(4,11)(5,12)(6,10)(8,15)(9,14)(13,16)(19,28)(21,30)(23,27)(25,31)(33,35,39,46)(34,40,44,43)(36,45,48,37)(38,41,42,47);(3,10)(7,14)(11,15)(12,16)(17,19,22,28)(18,23,26,27)(20,21,29,30)(24,31,32,25)(33,36,39,48)(34,41,44,47)(35,45,46,37)(38,40,42,43);(1,4)(2,5)(3,11)(6,8)(7,12)(9,13)(10,15)(14,16)(17,20)(18,24)(19,21)(22,29)(23,31)(25,27)(26,32)(28,30)(33,37)(34,42)(35,36)(38,44)(39,45)(40,47)(41,43)(46,48);(1,5)(2,8)(3,12)(4,9)(6,13)(7,15)(10,16)(11,14)(17,21)(18,25)(19,29)(20,28)(22,30)(23,24)(26,31)(27,32)(33,35)(34,43)(36,45)(37,48)(38,41)(39,46)(40,44)(42,47);(1,3)(2,7)(4,11)(5,12)(6,10)(8,15)(9,14)(13,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,32)(30,31)(33,38,39,42)(34,37,44,45)(35,41,46,47)(36,43,48,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,f^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, 8800629);