#500187

order	512
nilpotency class	4
isoclinism	isoclinic class of #499849
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	#46276	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1104 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,26)(14,28)(15,19)(18,30)(20,31)(21,36)(22,33)(23,39)(24,34)(25,35)(27,32)(29,37)(38,40);(2,5)(6,8)(9,11,15,13)(10,16,19,26)(12,22,25,38)(14,27,30,21)(17,33,35,40)(18,36,28,32)(20,29,23,24)(31,37,39,34);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,20)(13,23)(14,29)(15,25)(16,31)(18,37)(19,35)(21,38)(22,27)(24,30)(26,39)(28,34)(32,40)(33,36);(2,5)(6,8)(9,13,15,11)(10,16,19,26)(12,23,25,20)(14,27,30,21)(17,31,35,39)(18,32,28,36)(22,24,38,29)(33,37,40,34)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,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, #72311110, #72311115, …
maximal normal subgroup of	#72311092, #72311110, #72311115, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407383);