#371407

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
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	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1071 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:

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

Polycyclic presentation:

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

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, 278603);