#107101

order	512
nilpotency class	3
isoclinism	isoclinic class of #9385
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

889 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)(9,10,14,18,15,19,13,27)(11,20,22,16,23,32,28,31)(12,17,25,35,26,36,29,34)(21,38,40,33,24,37,39,30);(1,2,4,8,5,7,3,6)(9,11)(10,16)(12,24)(13,28)(14,22)(15,23)(17,33)(18,32)(19,31)(20,27)(21,26)(25,39)(29,40)(30,36)(34,38)(35,37);(2,7)(6,8)(9,12)(10,17)(11,21)(13,29)(14,25)(15,26)(16,30)(18,35)(19,36)(20,37)(22,40)(23,24)(27,34)(28,39)(31,33)(32,38)

Polycyclic presentation:

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

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, 14297);