#147399

order	512
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₄×D₈
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

738 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)(9,10,13,20,16,21,15,31)(11,22,25,35,28,40,27,33)(12,18,29,37,30,38,14,19)(17,32,34,23,36,26,39,24);(2,4)(9,11,14,26,15,27,30,23,16,28,29,32,13,25,12,24)(10,17,18,35,20,34,37,40,21,36,38,33,31,39,19,22);(5,6,7,8)(9,12,13,29,16,30,15,14)(10,18,20,37,21,38,31,19)(11,23,25,26,28,24,27,32)(17,33,34,22,36,35,39,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,c^b=ci

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287496, #72287546, #72287553
maximal normal subgroup of	#72287496, #72287546, #72287553

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54595);