#462911

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

686 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)(3,6,4,8)(9,10,16,30,14,20,15,21)(11,17,29,38,24,33,25,34)(12,19,28,36,13,18,27,37)(22,39,23,35,40,31,26,32);(1,3,5,4)(2,6,7,8)(9,11,14,24)(10,17,20,33)(12,26,13,23)(15,29,16,25)(18,35,19,32)(21,38,30,34)(22,28,40,27)(31,37,39,36);(2,7)(6,8)(9,12)(10,18)(11,22)(13,14)(15,28)(16,27)(17,31)(19,20)(21,37)(23,25)(24,40)(26,29)(30,36)(32,34)(33,39)(35,38);(1,4,5,3)(2,8,7,6)(9,13,14,12)(10,19)(11,23,24,26)(15,28,16,27)(17,32)(18,20)(21,37)(22,29,40,25)(30,36)(31,34)(33,35)(38,39)

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,d^2=fi,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,h^c=hi,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, 370107);