#147428

order	512
nilpotency class	4
isoclinism	isoclinic class of #12087
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#442
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

712 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)(3,4)(6,8)(9,10)(11,21,24,35)(12,27)(13,17)(14,30)(15,32)(16,23,34,25)(18,29)(19,31)(20,28)(22,38,26,36)(33,39,37,40);(2,4)(9,11,14,25,15,26,31,39,13,24,12,23,28,22,29,40)(10,16,19,35,20,33,30,36,17,34,18,21,32,37,27,38);(5,6,7,8)(10,17)(11,22,24,26)(12,14)(16,33,34,37)(20,32)(21,36,35,38)(23,40,25,39)(29,31)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,c^b=ch,f^b=fi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54624);