#474211

order	512
nilpotency class	3
isoclinism	isoclinic class of #461203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

542 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)(3,5)(4,6)(7,8)(9,10,15,21,16,22,14,20)(11,23,12,26,25,18,28,17)(13,19,31,36,32,37,30,35)(24,38,27,39,29,34,40,33);(1,3,4,7)(2,5,6,8)(9,11,16,25)(10,17)(12,15,28,14)(13,24,32,29)(18,21)(19,33)(20,23)(22,26)(27,31,40,30)(34,36)(35,38)(37,39);(9,12,15,25,16,28,14,11)(10,18,20,26,22,23,21,17)(13,29,31,40,32,24,30,27)(19,33,35,34,37,39,36,38);(2,6)(5,8)(9,13,15,31,16,32,14,30)(10,19,21,36,22,37,20,35)(11,24,28,40,25,29,12,27)(17,33,18,34,26,39,23,38)

Polycyclic presentation:

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

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, 381407);