#6948303

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1259 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)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26,31,35,30,34,29,33)(27,28,36,38,37,32,39,40);(1,3,7,4)(2,5,8,6)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,37)(26,32,34,28)(29,39,31,36)(33,38,35,40);(1,4,7,3)(2,5,8,6)(10,15)(11,17)(12,19)(20,24)(25,28)(26,27)(29,38)(30,32)(31,40)(33,36)(34,37)(35,39);(1,5,7,6)(2,3,8,4)(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(25,27)(26,32)(28,34)(29,39)(30,37)(31,36)(33,38)(35,40);(1,6,7,5)(2,3,8,4)(11,17)(14,21)(18,23)(20,24)(25,29,30,31)(26,33,34,35)(27,36,37,39)(28,38,32,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=fi,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,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, 6855499);