#8701678

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

3877 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8608874);