#149300

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1955
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

508 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)(3,8)(4,6)(5,7)(9,10,15,20,13,18,29,17,14,19,30,38,22,32,12,28)(11,21,25,37,26,16,23,35,27,36,31,33,24,34,40,39);(1,3,5,4)(2,6,7,8)(9,11,22,24,14,27,13,26)(10,16,32,21,19,34,18,36)(12,23,30,25,29,40,15,31)(17,37,20,39,28,33,38,35);(1,4,5,3)(2,6,7,8)(9,12,22,30,14,29,13,15)(10,17,32,20,19,28,18,38)(11,23,24,25,27,40,26,31)(16,33,21,35,34,37,36,39)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56496);