#8678989

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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)(3,12)(4,11)(5,10)(6,8)(7,9)(13,15)(14,16)(17,18,22,26)(19,23,20,24)(21,25,30,29)(27,31,28,32)(35,43)(37,47)(40,44)(42,48);(1,3,8,7)(2,9,6,12)(4,13,5,14)(10,15,11,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(33,34,39,38)(35,44,43,40)(36,46,45,41)(37,42,47,48);(1,4,8,5)(2,10,6,11)(3,13,7,14)(9,15,12,16)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,35,39,43)(34,40,38,44)(36,37,45,47)(41,42,46,48);(1,5,8,4)(2,11,6,10)(3,13,7,14)(9,15,12,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,36,39,45)(34,41,38,46)(35,37,43,47)(40,42,44,48);(1,6,8,2)(3,12,7,9)(4,11,5,10)(13,16,14,15)(17,20,22,19)(18,24,26,23)(21,27,30,28)(25,31,29,32)(33,37)(34,42)(35,45)(36,43)(38,48)(39,47)(40,46)(41,44);(1,7,8,3)(2,9,6,12)(4,14,5,13)(10,15,11,16)(17,21,22,30)(18,25,26,29)(19,27,20,28)(23,31,24,32)(33,38,39,34)(35,44,43,40)(36,41,45,46)(37,42,47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8586185);