#8096941

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	#45732	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4622 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)(9,10,16,18)(11,13,14,12)(15,17,23,24)(19,21,22,20)(25,26,30,34)(27,29,33,31)(28,32,36,39)(35,37,40,38);(2,5)(3,7)(9,11,16,14)(10,12,18,13)(15,19,23,22)(17,20,24,21)(28,36)(32,39)(35,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,14,18,11)(15,20,23,21)(17,22,24,19)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,13,16,12)(10,11,18,14)(15,21,23,20)(17,19,24,22)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(9,14,16,11)(10,12,18,13)(15,19,23,22)(17,21,24,20)(25,27)(26,29)(28,35)(30,33)(31,34)(32,37)(36,40)(38,39);(9,15)(10,17)(11,19)(12,20)(13,21)(14,22)(16,23)(18,24)(25,29)(26,33)(27,34)(28,37)(30,31)(32,40)(35,39)(36,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004137);