#8757339

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	#56210	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664535);