#226955

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	C₂×#454	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dei,e^b=eh,d^c=dh,f^c=fh,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134151);