#107769

order	512
nilpotency class	3
isoclinism	isoclinic class of #8528
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

770 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh,f^c=fhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14965);