#111182

order	512
nilpotency class	3
isoclinism	isoclinic class of #9239
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

795 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342986, #72342990, #72342997, …
maximal normal subgroup of	#72342986, #72342990, #72342997, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18378);