#8677404

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4446 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8584600);