#462612

order	512
nilpotency class	3
isoclinism	isoclinic class of #458129
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

672 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=e,c^2=i,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,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, 369808);