#8425345

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	#52317	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)(18,24)(19,20)(21,29)(30,32)(33,34,39,40)(35,44,45,41)(36,48,42,37)(38,43,46,47);(3,10)(7,8)(11,16)(12,15)(17,18)(19,25)(20,22)(21,26)(23,24)(27,32)(28,30)(29,31)(34,40)(35,45)(36,42)(43,47);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,15)(14,16)(17,19,23,20)(18,25,24,22)(21,27,29,28)(26,32,31,30)(33,35)(34,41)(36,46)(37,47)(38,42)(39,45)(40,44)(43,48);(1,3)(2,8)(4,11)(5,15)(6,10)(7,9)(12,13)(14,16)(17,20,23,19)(18,25,24,22)(21,28,29,27)(26,32,31,30)(33,36,39,42)(34,37,40,48)(35,46,45,38)(41,47,44,43);(1,4)(2,5)(3,11)(6,14)(7,12)(8,15)(9,13)(10,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,37)(34,42)(35,47)(36,40)(38,41)(39,48)(43,45)(44,46);(1,5)(2,4)(3,12)(6,13)(7,11)(8,16)(9,14)(10,15)(17,22,23,25)(18,20,24,19)(21,30,29,32)(26,28,31,27)(33,38,39,46)(34,43,40,47)(35,42,45,36)(37,41,48,44)

Polycyclic presentation:

2^9|a^2=i,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, 8332541);