#469146

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
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)(13,29,32,37)(14,21)(15,20)(16,22)(17,25)(18,26)(19,31,38,33)(24,34,30,36)(28,39,40,35);(1,2,4,5)(3,7,8,6)(9,11,14,25,16,26,15,12)(10,17,20,23,22,27,21,18)(13,24,33,40,32,30,31,28)(19,34,29,39,38,36,37,35);(1,3)(2,6)(4,8)(5,7)(9,12,15,26,16,25,14,11)(10,18,21,27,22,23,20,17)(13,30,33,28,32,24,31,40)(19,36,29,35,38,34,37,39);(3,8)(6,7)(9,13,15,31,16,32,14,33)(10,19,21,37,22,38,20,29)(11,24,12,28,26,30,25,40)(17,34,18,35,27,36,23,39)

Polycyclic presentation:

2^9|a^2=efg,b^2=fgi,c^2=egi,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,h^b=hi,d^c=de,h^c=hi,e^d=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, 376342);