#149038

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₄×D₈
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

490 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:

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

Polycyclic presentation:

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

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, 56234);