#6380548

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,d^b=df,f^b=fi,e^c=ei,f^d=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287744);