#484807

order	512
nilpotency class	4
isoclinism	isoclinic class of #484606
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	C₂×C₄⋊C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

992 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi

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, 392003);