#369060

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1908 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fg,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276256);