#470595

order	512
nilpotency class	3
isoclinism	isoclinic class of #454681
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)(9,10,13,18,14,16,26,30)(11,19,22,32,23,15,20,27)(12,17,25,31,24,29,37,40)(21,33,36,39,35,28,34,38);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,21)(13,20)(14,23)(16,19)(17,28)(18,32)(22,26)(24,35)(25,34)(27,30)(29,33)(31,39)(36,37)(38,40);(10,16)(11,20)(13,26)(15,27)(19,32)(21,34)(22,23)(25,37)(28,39)(31,40)(33,38)(35,36);(2,6)(5,8)(9,12)(10,17)(11,21)(13,25)(14,24)(15,28)(16,29)(18,31)(19,33)(20,34)(22,36)(23,35)(26,37)(27,38)(30,40)(32,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=de,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, 377791);