#8467138

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4171 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=i,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8374334);