#443513

order	512
nilpotency class	3
isoclinism	isoclinic class of #420420
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
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

1102 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 350709);