#469150

order	512
nilpotency class	3
isoclinism	isoclinic class of #455826
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

821 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)(6,7)(9,10)(11,23)(12,27)(13,19,31,37)(14,21)(15,20)(16,22)(17,25)(18,26)(24,34,29,39)(28,33,40,35)(30,38,32,36);(1,2,4,5)(3,7,8,6)(9,11,14,25,16,26,15,12)(10,17,20,23,22,27,21,18)(13,24,32,40,31,29,30,28)(19,33,38,39,37,35,36,34);(1,3)(2,6)(4,8)(5,7)(9,12,15,26,16,25,14,11)(10,18,21,27,22,23,20,17)(13,29,32,28,31,24,30,40)(19,35,38,34,37,33,36,39);(3,8)(6,7)(9,13,15,30,16,31,14,32)(10,19,21,36,22,37,20,38)(11,24,12,28,26,29,25,40)(17,33,18,34,27,35,23,39)

Polycyclic presentation:

2^9|a^2=efg,b^2=fgi,c^2=egi,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,h^b=hi,d^c=de,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, 376346);