#371411

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1091 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278607);