#8339876

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	#2675	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4309 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,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, 8247072);