#6374214

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342674
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#3485	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,h^c=hi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284639, #72284662, #72284740, …
maximal normal subgroup of	#72284639, #72284662, #72284740, …
maximal quotient of	#72314039

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281410);