#6937126

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	#2433	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=fi,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6844322);