#497652

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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#554 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

882 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,d^2=ei,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, 404848);