#7181993

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	#60924	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1681 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,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, 7089189);