#370952

order	512
nilpotency class	3
isoclinism	isoclinic class of #23906
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₄×#120
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1268 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,e^c=ei,f^c=fi,h^c=hi,f^d=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, 278148);