#6986055

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907961
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 ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1311 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)(3,8)(4,7)(5,6)(9,10,15,18)(11,19,12,16)(13,22,14,17)(20,23,21,24)(25,26)(27,33)(28,36)(29,31)(30,34)(32,37)(35,38)(39,40);(2,6)(3,7)(9,11,15,12)(10,16,18,19)(13,20,14,21)(17,23,22,24)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(1,3,8,2,5,7,4,6)(9,12,15,11)(10,16,18,19)(13,20,14,21)(17,24,22,23)(26,31)(27,34)(32,36)(35,40);(2,6)(3,7)(9,13,15,14)(10,17,18,22)(11,20,12,21)(16,23,19,24)(25,28,29,37)(26,32,31,36)(27,35,34,40)(30,38,33,39);(1,4,5,8)(2,7,6,3)(9,14,15,13)(10,17,18,22)(11,20,12,21)(16,24,19,23)

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=fg,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfhi,e^c=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, 6893251);