#8621819

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	#48105	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=h,e^2=hi,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, 8529015);