#6947904

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884873
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	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1388 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:

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

Polycyclic presentation:

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

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, 6855100);