#373777

order	512
nilpotency class	3
isoclinism	isoclinic class of #23906
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1100 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280973);