#468774

order	512
nilpotency class	3
isoclinism	isoclinic class of #459488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

750 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)(9,10,13,19,14,20,15,17)(11,21,24,16,22,31,25,32)(12,26,27,18,28,34,29,33)(23,37,39,35,36,30,38,40);(2,5)(6,8)(9,11)(10,16,20,32)(12,23)(13,24)(14,22)(15,25)(17,21,19,31)(18,30,33,37)(26,35,34,40)(27,39)(28,36)(29,38);(1,3,4,7)(2,6,5,8)(10,17)(11,22)(12,27)(13,15)(16,21)(18,33)(19,20)(23,38)(28,29)(30,37)(31,32)(36,39);(3,7)(6,8)(9,12,13,27,14,28,15,29)(10,18,19,34,20,33,17,26)(11,23,24,39,22,36,25,38)(16,30,31,40,32,37,21,35)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=efg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,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, 375970);