#108164

order	512
nilpotency class	3
isoclinism	isoclinic class of #8479
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

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, 15360);