#497517

order	512
nilpotency class	4
isoclinism	isoclinic class of #480866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#546 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1034 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311502, #72311505, #72311510
maximal normal subgroup of	#72311502, #72311505, #72311510

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404713);