#369361

order	512
nilpotency class	3
isoclinism	isoclinic class of #196299
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1452 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=efg,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,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, 276557);