#8095555

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	C₂×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4374 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002751);