#470910

order	512
nilpotency class	3
isoclinism	isoclinic class of #460861
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

593 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)(4,8)(5,7)(9,10)(11,15,22,30)(12,16)(13,19)(14,17)(18,20)(21,28,24,32)(23,29,36,31)(25,34)(26,35)(27,33)(37,40,38,39);(1,3,5,4)(2,6,7,8)(9,11,20,36,13,22,14,23)(10,15,18,31,19,30,17,29)(12,24,27,37,25,21,26,38)(16,32,33,40,34,28,35,39);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,21)(13,25)(14,27)(15,28)(17,33)(18,35)(19,34)(20,26)(22,24)(23,38)(29,39)(30,32)(31,40)(36,37);(10,17,19,18)(12,25)(15,29,30,31)(16,33,34,35)(21,24)(26,27)(28,39,32,40)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=efh,c^2=f,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,d^c=de,h^c=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, 378106);