#497525

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×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#546 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1060 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)(11,20)(12,17)(13,18)(14,19,27,16)(15,21)(22,32)(23,30,35,28)(24,29)(25,33)(26,34,38,31)(36,40,37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,24)(13,21)(14,23)(16,30)(17,32)(18,20)(19,28)(22,25)(26,37)(27,35)(29,33)(31,39)(34,40)(36,38);(1,3,4,7)(2,6,5,8)(10,16)(11,21)(14,27)(15,28)(17,31)(18,19)(20,30)(22,24)(26,38)(29,39)(32,40)(33,34);(2,5)(6,8)(9,12,13,25)(10,17,18,33)(11,22,21,24)(14,26,27,38)(15,29,20,32)(16,31,19,34)(23,36,35,37)(28,39,30,40)

Polycyclic presentation:

2^9|a^2=g,c^2=ei,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, 404721);