#116311

order	512
nilpotency class	3
isoclinism	isoclinic class of #10320
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

766 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)(9,10,13,18,14,17,25,31)(11,15,21,27,20,28,33,38)(12,16,24,30,23,29,36,39)(19,26,32,37,22,34,35,40);(1,2)(3,5)(4,7)(6,8)(9,11)(10,15)(12,22,23,19)(13,21)(14,20)(16,26,29,34)(17,28)(18,27)(24,35,36,32)(25,33)(30,37,39,40)(31,38);(2,5)(3,8)(6,7)(9,12)(10,16)(11,19)(13,24)(14,23)(15,26)(17,29)(18,30)(20,22)(21,32)(25,36)(27,37)(28,34)(31,39)(33,35)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,e^b=ei,f^b=fhi,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224908, #72224912, #72224913, …
maximal normal subgroup of	#72224908, #72224912, #72224913, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23507);