#470594

order	512
nilpotency class	3
isoclinism	isoclinic class of #454664
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377790);