#139740

order	512
nilpotency class	4
isoclinism	isoclinic class of #139739
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

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

normal subgroups only

This group has 678 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=ehi,f^b=fg,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	#72209847
maximal normal subgroup of	#72209847
maximal quotient of	#72227676, #72227719, #72227724, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46936);