#8697561

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

4250 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8604757);