#121015

order	512
nilpotency class	3
isoclinism	isoclinic class of #121012
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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)(3,13,15,8,17,24,16,25)(4,18,20,28,21,27,22,9)(14,26,31,32,19,23,30,29)(33,34)(35,39,44,51)(36,40,47,54)(37,42)(38,41)(43,50)(45,52,58,63)(46,53)(48,55,60,61)(49,56)(57,62)(59,64);(1,3,5,16,7,17,6,15)(2,8,11,13,12,25,10,24)(4,19,20,30,21,14,22,31)(9,26,18,32,28,23,27,29)(33,35)(34,39,41,51)(36,46,47,59)(37,45)(38,44)(40,53)(42,52,56,63)(43,48,57,60)(49,58)(50,61)(54,64)(55,62);(1,4,5,20,7,21,6,22)(2,9,10,27,12,28,11,18)(3,14,15,30,17,19,16,31)(8,23,13,29,25,26,24,32)(33,36)(34,40,41,54)(35,43,44,57)(37,48)(38,47)(39,50)(42,55,56,61)(45,46,58,59)(49,60)(51,62)(52,53)(63,64)

Polycyclic presentation:

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

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, 28211);