#8167428

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8074624);