#467999

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

628 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:

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

Polycyclic presentation:

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

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, 375195);