#470778

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

613 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,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, 377974);