#8429379

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	#53066	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=g,e^2=g,f^2=gi,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, 8336575);