#454967

order	512
nilpotency class	3
isoclinism	isoclinic class of #454039
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60426	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1485 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)(3,6)(4,5)(7,8)(9,10,14,19,16,20,15,31)(11,17,23,34,25,36,24,35)(12,26,28,37,13,29,30,18)(21,38,22,40,39,33,27,32);(2,5)(6,8)(9,11)(10,17)(12,27)(13,22)(14,23)(15,24)(16,25)(18,33)(19,34)(20,36)(21,28)(26,32)(29,40)(30,39)(31,35)(37,38);(3,7)(6,8)(9,12,16,13)(10,18,20,37)(11,21,25,39)(14,30,15,28)(17,32,36,40)(19,29,31,26)(22,23,27,24)(33,35,38,34);(1,3,4,7)(2,6,5,8)(9,13,16,12)(10,18,20,37)(11,22,25,27)(14,30,15,28)(17,33,36,38)(19,26,31,29)(21,23,39,24)(32,35,40,34)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,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, 362163);