#496514

order	512
nilpotency class	4
isoclinism	isoclinic class of #480662
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

939 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311122, #72311129, #72311186, …
maximal normal subgroup of	#72311122, #72311129, #72311186, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403710);