#139744

order	512
nilpotency class	4
isoclinism	isoclinic class of #139743
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9495
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#23896	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

668 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dgi,e^b=ehi,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,f^c=fh,g^c=gi,e^d=ei,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342824
maximal normal subgroup of	#72342824

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46940);