#127188

order	512
nilpotency class	4
isoclinism	isoclinic class of #7213
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₂×#342
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

701 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gh,f^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,e^b=eh,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240721, #72240805, #72240806, …
maximal normal subgroup of	#72240721, #72240805, #72240806, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34384);