#468000

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

Polycyclic presentation:

2^9|a^2=efg,b^2=g,c^2=fi,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, 375196);