#500040

order	512
nilpotency class	4
isoclinism	isoclinic class of #499831
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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	#32899	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1604 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)(9,10,13,15)(11,16,21,28)(12,22,23,17)(14,26,25,18)(19,31,27,29)(20,34,36,30)(24,32,38,37)(33,39,35,40);(1,3,4,8)(2,5,6,7)(10,15)(11,19)(12,23)(14,25)(16,29)(20,35)(21,27)(28,31)(30,40)(32,37)(33,36)(34,39);(9,11)(10,16)(12,20)(13,21)(14,27)(15,28)(17,30)(18,29)(19,25)(22,34)(23,36)(24,35)(26,31)(32,39)(33,38)(37,40);(2,6)(5,7)(9,12,13,23)(10,17,15,22)(11,20,21,36)(14,24,25,38)(16,30,28,34)(18,32,26,37)(19,33,27,35)(29,39,31,40)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311092, #72311104, #72311122, …
maximal normal subgroup of	#72311092, #72311104, #72311122, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407236);