#8701512

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

3871 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(2,3)(7,9)(8,10)(15,16)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,30)(29,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,18)(19,25,27,23)(20,24,29,28)(26,30,32,31)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=ghi,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, 8608708);