#128947

order	512
nilpotency class	4
isoclinism	isoclinic class of #11989
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1551
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8235
prime radical	C₂³	#413
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

283 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=g,e^2=i,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fgh,h^a=hi,c^b=ch,d^b=dh,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265701, #72265714, #72265723
maximal normal subgroup of	#72265701, #72265714, #72265723

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36143);