#8750808

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

4028 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=gi,e^2=h,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, 8658004);