#6532504

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=hi,d^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6439700);