#146740

order	512
nilpotency class	4
isoclinism	isoclinic class of #128584
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10788
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

955 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dhi,e^b=ehi,h^b=hi,d^c=dhi,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279896, #72280442, #72280507, …
maximal normal subgroup of	#72279896, #72280442, #72280507, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53936);