#8455705

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	#56438	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4354 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)(3,7)(4,6)(5,8)(9,10,16,14)(11,21,23,17)(12,19,13,18)(15,20,24,22)(25,26,31,32)(27,36,37,33)(28,40,34,29)(30,35,38,39);(1,3)(2,5)(4,8)(6,7)(9,11,16,23)(10,17,14,21)(12,22,13,20)(15,19,24,18)(26,32)(27,37)(28,34)(35,39);(9,12,16,13)(10,18,14,19)(11,22,23,20)(15,21,24,17)(25,27)(26,33)(28,38)(29,39)(30,34)(31,37)(32,36)(35,40);(2,6)(5,7)(9,13,16,12)(10,19,14,18)(11,22,23,20)(15,21,24,17)(25,28,31,34)(26,29,32,40)(27,38,37,30)(33,39,36,35);(1,3)(2,5)(4,8)(6,7)(9,14,16,10)(11,21,23,17)(12,19,13,18)(15,22,24,20)(25,29)(26,34)(27,39)(28,32)(30,33)(31,40)(35,37)(36,38);(1,2)(3,5)(4,6)(7,8)(9,15,16,24)(10,20,14,22)(11,19,23,18)(12,21,13,17)(25,30,31,38)(26,35,32,39)(27,34,37,28)(29,33,40,36)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8362901);