#121016

order	512
nilpotency class	3
isoclinism	isoclinic class of #120689
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 355 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=hi,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=dh,f^b=fh,g^b=gi,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28212);