#8634316

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	#37954	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4094 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,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,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, 8541512);