#500989

order	512
nilpotency class	4
isoclinism	isoclinic class of #499803
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	C₂×H(5,𝔽₂)	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

894 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408185);