#8425339

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	#59689	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4157 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)(3,8,10,7)(4,13,14,5)(11,12,16,15)(18,20)(24,25)(26,28)(31,32)(33,34)(35,38)(36,46)(37,45)(39,44)(40,43)(41,48)(42,47);(3,10)(7,8)(11,16)(12,15)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(33,35,39,43)(34,40,44,38)(36,45,48,42)(37,46,47,41);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,15)(14,16)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,36)(34,41)(35,45)(37,40)(38,47)(39,48)(42,43)(44,46);(1,3)(2,8)(4,11)(5,15)(6,10)(7,9)(12,13)(14,16)(17,20,23,18)(19,25,21,24)(22,26,30,28)(27,31,29,32)(33,37,39,47)(34,42,44,45)(35,41,43,46)(36,40,48,38);(1,4)(2,5)(3,11)(6,14)(7,12)(8,15)(9,13)(10,16)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,38,39,40)(34,43,44,35)(36,47,48,37)(41,42,46,45);(1,5)(2,4)(3,12)(6,13)(7,11)(8,16)(9,14)(10,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,34)(35,40)(36,41)(37,45)(38,43)(39,44)(42,47)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332535);