#116754

order	512
nilpotency class	3
isoclinism	isoclinic class of #10238
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

717 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23950);