#371327

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1257 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,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, 278523);