#121239

order	512
nilpotency class	3
isoclinism	isoclinic class of #121238
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

350 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28435);