#8634417

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	#39144	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=gh,e^2=g,f^2=gi,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, 8541613);