#497527

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#547 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

994 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404723);