#497532

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂×#553	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×D₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

966 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)(9,10,14,19,12,17,25,33)(11,20,23,35,21,15,28,30)(13,18,26,34,24,16,27,32)(22,31,37,40,36,29,38,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,21)(13,22)(14,23)(16,31)(17,20)(18,29)(19,30)(24,36)(25,28)(26,37)(27,38)(32,40)(33,35)(34,39);(1,3,4,7)(2,6,5,8)(10,16)(11,21)(13,24)(15,29)(17,18)(19,34)(20,31)(23,28)(26,27)(30,40)(32,33)(35,39);(2,5)(6,8)(11,21)(15,20)(22,36)(23,28)(29,31)(30,35)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=g,c^2=ei,f^2=i,g^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, 404728);