#8167432

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	#40365	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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)(17,18,23,21)(19,26,22,24)(20,28,29,25)(27,31,30,32)(33,34,39,40)(35,37,36,41)(38,42,46,47)(43,45,44,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,23,22)(18,24,21,26)(20,27,29,30)(25,31,28,32)(34,40)(35,36)(42,47)(43,44);(2,3)(7,9)(8,10)(14,15)(20,29)(25,28)(27,30)(31,32)(33,35,39,36)(34,41,40,37)(38,43,46,44)(42,48,47,45);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32)(33,36,39,35)(34,37,40,41)(38,44,46,43)(42,45,47,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,18)(19,26,22,24)(20,25,29,28)(27,31,30,32)(33,37)(34,35)(36,40)(38,45)(39,41)(42,43)(44,47)(46,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22,23,19)(18,26,21,24)(20,30,29,27)(25,32,28,31)(33,38,39,46)(34,42,40,47)(35,43,36,44)(37,45,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=i,e^2=gi,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, 8074628);