#141262

order	512
nilpotency class	4
isoclinism	isoclinic class of #12436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1667
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9734
prime radical	#33713	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

349 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 349 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,13,7,14,28,30,8,15,29,41,6,11,23,12)(3,9,19,33,20,35,18,42,21,36,44,32,17,34,50,60)(4,22,24,10,26,38,54,39,27,52,55,40,25,51,53,37)(16,43,47,57,48,62,46,31,49,56,63,58,45,61,64,59);(1,3,5,18,7,17,28,19,8,21,29,50,6,20,23,44)(2,9,30,33,11,35,13,42,15,36,12,32,14,34,41,60)(4,16,24,46,26,45,54,47,27,49,55,64,25,48,53,63)(10,31,52,56,37,58,38,61,40,59,22,43,39,57,51,62);(1,4,23,53,6,25,29,55,8,27,28,54,7,26,5,24)(2,10,13,38,14,39,30,52,15,40,41,51,11,37,12,22)(3,16,44,63,20,48,50,64,21,49,19,47,17,45,18,46)(9,31,42,61,34,57,33,56,36,59,60,62,35,58,32,43)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=eg,d^2=gh,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225389, #72225399, #72225406, …
maximal normal subgroup of	#72225389, #72225399, #72225406, …
maximal quotient of	#72225477

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48458);