#8110529

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	#45687	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4072 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=i,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017725);