#502714

order	512
nilpotency class	4
isoclinism	isoclinic class of #501172
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#3485	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

944 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^c=gh,h^c=hi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409910);