#120965

order	512
nilpotency class	3
isoclinism	isoclinic class of #10645
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

358 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 116 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)(3,13,16,24,17,27,15,8)(4,18,21,26,22,28,20,9)(14,30,31,25,32,23,19,29)(33,34)(35,44)(36,49)(37,53)(38,42)(39,43)(40,46)(41,51)(45,57)(47,56)(48,61)(50,60)(52,58)(54,59)(55,63)(62,64);(1,3,5,15,7,17,6,16)(2,8,10,13,12,24,11,27)(4,19,20,32,22,31,21,14)(9,25,18,23,26,29,28,30)(33,35,37,47,39,46,38,48)(34,40,42,56,43,44,53,61)(36,50,51,62,52,63,54,45)(41,57,49,55,59,64,58,60);(1,4,5,20,7,22,6,21)(2,9,10,18,12,26,11,28)(3,14,15,19,17,32,16,31)(8,23,13,29,24,30,27,25)(33,36)(34,41)(35,45,46,62)(37,54)(38,51)(39,52)(40,55,44,60)(42,58)(43,59)(47,63,48,50)(49,53)(56,57,61,64)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28161);