#8425663

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	#40947	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4032 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=gh,e^2=gh,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, 8332859);