#121752

order	512
nilpotency class	3
isoclinism	isoclinic class of #11000
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

516 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28948);