#454496

order	512
nilpotency class	3
isoclinism	isoclinic class of #454021
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
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

1467 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,f^c=fi,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, 361692);