#470852

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

739 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)(9,10,14,22,15,21,16,20)(11,23,26,17,28,36,27,37)(12,18,31,38,32,25,30,39)(13,24,33,29,34,40,35,19);(2,5)(6,8)(9,11,16,27,15,28,14,26)(10,17,22,36,21,37,20,23)(12,29,31,40,32,19,30,24)(13,25,35,38,34,18,33,39);(2,5)(3,7)(9,12,16,30,15,32,14,31)(10,18,22,38,21,25,20,39)(11,24,26,29,28,40,27,19)(13,23,35,37,34,36,33,17);(1,3,4,7)(2,6,5,8)(9,13,14,33,15,34,16,35)(10,19,20,40,21,29,22,24)(11,25,26,39,28,18,27,38)(12,17,30,23,32,37,31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378048);