#107777

order	512
nilpotency class	3
isoclinism	isoclinic class of #9307
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

883 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)(3,4)(7,8)(9,10)(11,20)(12,24,27,35)(13,28)(14,18)(15,19)(16,22)(17,26,36,30)(21,34,25,31)(23,32)(29,33)(37,39,40,38);(1,2,4,8,5,7,3,6)(9,11)(10,16)(12,25)(13,29)(14,22)(15,23)(17,34)(18,32)(19,33)(20,28)(21,27)(24,40)(26,38)(30,39)(31,36)(35,37);(2,7)(6,8)(9,12)(10,17)(11,21)(13,26)(14,30)(15,27)(16,31)(18,35)(19,36)(20,37)(22,38)(23,25)(24,28)(29,39)(32,40)(33,34)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,c^b=ch,d^b=di,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, 14973);