#128932

order	512
nilpotency class	4
isoclinism	isoclinic class of #11989
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×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	#8234
prime radical	#3147	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 369 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=i,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,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, #72265710, #72265723
maximal normal subgroup of	#72265701, #72265710, #72265723

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36128);