#8429415

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 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)(3,7)(4,12)(5,9)(6,10)(8,14)(11,13)(15,16)(18,20)(19,25)(24,28)(27,29)(33,34,38,37)(35,39,41,43)(36,40,44,45)(42,46,47,48);(1,3,6,9)(2,7,10,5)(4,11,14,16)(8,15,12,13)(17,18,22,20)(19,26,25,23)(21,24,30,28)(27,32,29,31)(35,41)(39,43)(42,47)(46,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,15)(14,16)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32)(33,35,38,41)(34,39,37,43)(36,42,44,47)(40,46,45,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,13)(14,16)(17,19,22,25)(18,23,20,26)(21,29,30,27)(24,32,28,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,4,6,14)(2,8,10,12)(3,11,9,16)(5,13,7,15)(17,20,22,18)(19,26,25,23)(21,28,30,24)(27,32,29,31);(1,5,6,7)(2,9,10,3)(4,13,14,15)(8,16,12,11)(17,21,22,30)(18,24,20,28)(19,27,25,29)(23,31,26,32)(33,37,38,34)(35,43,41,39)(36,40,44,45)(42,46,47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=g,e^2=h,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, 8336611);