#8425372

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	#38082	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332568);