#8633810

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	#45912	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4105 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)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31)(33,34,39,41)(35,37,38,40)(36,44,47,42)(43,48,45,46);(1,3,7,9)(2,4,5,12)(6,10,15,16)(8,11,13,14)(17,19)(18,23)(20,25)(21,29)(22,27)(24,28)(26,30)(31,32)(33,35,39,38)(34,40,41,37)(36,45,47,43)(42,46,44,48);(2,5)(3,9)(8,13)(10,16)(21,30)(24,32)(26,29)(28,31)(34,41)(37,40)(42,44)(46,48);(1,4)(2,3)(5,9)(6,14)(7,12)(8,16)(10,13)(11,15)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(33,36,39,47)(34,42,41,44)(35,43,38,45)(37,46,40,48);(1,5,7,2)(3,4,9,12)(6,13,15,8)(10,11,16,14)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,37,39,40)(34,35,41,38)(36,46,47,48)(42,43,44,45);(1,6)(2,8)(3,10)(4,11)(5,13)(7,15)(9,16)(12,14)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,30)(31,32)(33,38,39,35)(34,40,41,37)(36,43,47,45)(42,46,44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=h,f^2=hi,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, 8541006);