#8621633

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	#45606	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8528829);