#7054919

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61071	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2066 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)(3,5)(4,6)(7,8)(9,10,16,22)(11,23,25,30)(12,26,29,36)(13,19,17,31)(14,21,15,20)(18,32,35,28)(24,33,39,27)(34,40,38,37);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,24)(13,22)(14,31)(15,19)(16,25)(18,33)(20,30)(21,23)(26,40)(27,35)(28,38)(29,39)(32,34)(36,37);(1,3,4,8)(2,5,6,7)(9,12,16,29)(10,18,22,35)(11,24,25,39)(13,27,17,33)(14,28,15,32)(19,34,31,38)(20,26,21,36)(23,37,30,40);(3,8)(5,7)(9,13)(10,19)(11,20)(12,27)(14,23)(15,30)(16,17)(18,34)(21,25)(22,31)(24,26)(28,37)(29,33)(32,40)(35,38)(36,39);(2,6)(5,7)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6962115);