#8750812

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658008);