#8919256

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	#51104	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4222 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8826452);