#111191

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

783 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)(9,10,13,18,14,17,26,30)(11,15,21,28,20,29,24,33)(12,16,23,31,22,32,25,34)(19,35,36,40,37,27,38,39);(2,5)(3,8)(6,7)(9,11)(10,15,17,29)(12,19)(13,24)(14,20)(16,27,32,35)(18,33,30,28)(21,26)(22,37)(23,38)(25,36)(31,40,34,39);(9,12,13,25,14,22,26,23)(10,16,30,31,17,32,18,34)(11,19,24,36,20,37,21,38)(15,27,28,40,29,35,33,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,f^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	#72342988, #72342990, #72342997, …
maximal normal subgroup of	#72342988, #72342990, #72342997, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18387);