#120822

order	512
nilpotency class	3
isoclinism	isoclinic class of #10374
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

505 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,f^b=fhi,g^b=ghi,e^c=eh

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, 28018);