#114910

order	512
nilpotency class	3
isoclinism	isoclinic class of #10326
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	#1438
prime radical	#60949	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

932 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22106);