#121240

order	512
nilpotency class	3
isoclinism	isoclinic class of #121238
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 330 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)(3,12,15,8)(4,16)(5,20)(6,11)(7,10)(9,18)(13,30,32,26)(14,24,21,25)(17,23,31,29)(19,28)(22,27)(33,34,38,43,39,42,37,52)(35,44,47,56,46,40,53,57)(36,48,50,59,51,41,54,60)(45,61,63,64,62,55,49,58);(1,3,5,14,7,15,6,21)(2,8,20,25,10,12,11,24)(4,17,18,13,19,31,22,32)(9,26,28,23,27,30,16,29)(33,35)(34,40,42,44)(36,49)(37,53)(38,47)(39,46)(41,58,48,64)(43,57,52,56)(45,50)(51,63)(54,62)(55,60,61,59);(1,4)(2,9,10,27)(3,13)(5,18)(6,22)(7,19)(8,23,12,29)(11,16,20,28)(14,31)(15,32)(17,21)(24,26,25,30)(33,36)(34,41)(35,45,46,62)(37,50)(38,54)(39,51)(40,55,44,61)(42,48)(43,59)(47,49,53,63)(52,60)(56,58,57,64)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,e^2=h,f^2=hi,g^2=i,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, 28436);