#451042

order	512
nilpotency class	3
isoclinism	isoclinic class of #417562
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1372 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)(3,6)(4,5)(7,8)(9,10)(11,20,23,15)(12,17,25,33)(13,16)(14,18)(19,27)(21,29)(22,30,26,28)(24,31,38,34)(32,36)(35,39)(37,40);(2,5)(3,7)(9,11)(10,15)(12,21)(13,26)(14,23)(16,30)(17,32)(18,20)(19,28)(22,27)(24,37)(25,36)(29,33)(31,39)(34,40)(35,38);(10,16)(13,27)(15,28)(17,31)(18,19)(20,30)(22,26)(24,38)(29,39)(32,40)(33,34)(35,37);(1,3,4,7)(2,6,5,8)(9,12,14,25)(10,17)(11,21,23,36)(13,24,27,38)(15,29)(16,31)(18,33)(19,34)(20,32)(22,35,26,37)(28,39)(30,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318058, #72318259, #72318305
maximal normal subgroup of	#72318058, #72318259, #72318305

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358238);