#369402

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂×#386
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1476 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,22,26,16)(13,18)(14,17)(19,35,37,28)(20,29)(21,30)(23,39,38,31)(24,34)(25,33,27,32)(36,40);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,23,26,38)(13,20)(14,21)(16,31,22,39)(17,30)(18,29)(19,25,37,27)(24,36)(28,32,35,33)(34,40);(1,3,5,4)(2,7,8,6)(9,12,24,25,13,26,14,27)(10,16,17,33,18,22,34,32)(11,19,36,23,20,37,21,38)(15,28,30,39,29,35,40,31);(1,4,5,3)(2,7,8,6)(11,20)(15,29)(19,37)(21,36)(23,38)(28,35)(30,40)(31,39)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=efg,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi

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, 276598);