#8425346

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	#37484	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4162 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)(17,18,23,28)(19,29,31,24)(20,26,21,25)(22,27,30,32)(33,34)(35,42)(36,39)(37,44)(38,41)(40,46)(43,45)(47,48);(3,10)(7,8)(11,16)(12,15)(17,19)(18,24)(20,30)(21,22)(23,31)(25,32)(26,27)(28,29)(34,36)(35,41)(40,44)(43,47);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,15)(14,16)(17,20,23,21)(18,25,28,26)(19,30,31,22)(24,32,29,27)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,3)(2,8)(4,11)(5,15)(6,10)(7,9)(12,13)(14,16)(17,21,23,20)(18,26,28,25)(19,30,31,22)(24,32,29,27)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,4)(2,5)(3,11)(6,14)(7,12)(8,15)(9,13)(10,16)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,31)(30,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48);(1,5)(2,4)(3,12)(6,13)(7,11)(8,16)(9,14)(10,15)(17,22,23,30)(18,27,28,32)(19,21,31,20)(24,26,29,25)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=g,f^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, 8332542);