#474232

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

646 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=f,f^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,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 381428);