#497031

order	512
nilpotency class	4
isoclinism	isoclinic class of #483993
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

575 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)(10,17)(11,23,27,39)(12,14)(13,15)(18,32,34,37)(19,36)(21,22)(24,33,35,40)(25,28,38,26)(29,30);(1,3)(2,5)(4,8)(6,7)(9,10,15,21,16,22,13,17)(11,24,26,34,27,35,28,18)(12,19,29,20,30,36,14,31)(23,37,38,33,39,32,25,40);(3,8)(5,7)(9,11,14,23,15,28,12,25,16,27,29,39,13,26,30,38)(10,18,20,33,21,35,36,37,22,34,31,40,17,24,19,32);(2,6)(5,7)(9,12,13,14,16,30,15,29)(10,19,17,31,22,36,21,20)(11,25,26,23,27,38,28,39)(18,32,24,40,34,37,35,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343229, #72343235, #72343252
maximal normal subgroup of	#72343229, #72343235, #72343252
maximal quotient of	#72343544, #72343573

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404227);