#8634523

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4031 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=g,e^2=h,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, 8541719);