#8188055

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4040 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8095251);