#470593

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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 377789);