#8701729

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

4153 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=gh,e^2=ghi,f^2=gi,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, 8608925);