#467130

order	512
nilpotency class	3
isoclinism	isoclinic class of #454968
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1229 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,d^2=ei,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,h^c=hi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268304
maximal normal subgroup of	#72268304
maximal quotient of	#72268314

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 374326);