#8339815

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	#37614	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8247011);