#496579

order	512
nilpotency class	4
isoclinism	isoclinic class of #480690
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×Q₈	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

876 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)(9,10,15,20,13,18,12,25)(11,21,22,37,23,16,26,32)(14,19,27,34,28,35,29,17)(24,36,40,33,39,31,38,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,26)(13,23)(14,24)(15,22)(17,33)(18,21)(19,31)(20,32)(25,37)(27,40)(28,39)(29,38)(30,34)(35,36);(2,5)(6,8)(9,12,13,15)(10,17,18,34)(11,22,23,26)(14,29,28,27)(16,30,21,33)(19,20,35,25)(24,40,39,38)(31,37,36,32);(1,3,4,7)(2,6,5,8)(10,18)(11,23)(17,34)(19,35)(20,25)(22,26)(24,39)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403775);