#6826412

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1221 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)(4,5)(7,8)(9,10,13,15)(11,17,19,14)(12,16,21,23)(18,24,20,22)(25,26)(27,32)(28,33)(29,39)(30,35)(31,36)(34,38)(37,40);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27)(26,32)(28,29)(30,38)(31,37)(33,39)(34,35)(36,40);(2,5)(6,8)(25,27,28,38,31,37,30,29)(26,32,33,34,36,40,35,39);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(25,28,31,30)(26,33,36,35)(27,29,37,38)(32,39,40,34);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,29,28,27,31,38,30,37)(26,34,33,40,36,39,35,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=hi,d^2=gi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6733608);