#370350

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1072 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:

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

Polycyclic presentation:

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

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, 277546);