#146796

order	512
nilpotency class	4
isoclinism	isoclinic class of #12750
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1786
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 ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

943 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)(3,16,20,46)(4,23,26,58)(5,29)(6,32)(7,13)(8,15)(9,18,37,53)(10,25,40,60)(11,30)(12,31)(14,34)(17,48)(19,43,54,38)(21,47,22,36)(24,44,33,41)(27,42,28,39)(35,56)(45,59)(49,61)(50,64)(51,63)(52,62)(55,57);(1,3,7,21,8,20,34,22)(2,9,13,38,15,37,14,43)(4,17,27,51,26,52,28,56)(5,19,30,53,31,54,6,18)(10,35,41,48,40,63,44,62)(11,36,12,16,32,47,29,46)(23,57,42,45,58,61,39,64)(24,59,60,49,33,50,25,55);(1,4)(2,10)(3,17)(5,25)(6,33)(7,27)(8,26)(9,35)(11,23)(12,42)(13,41)(14,44)(15,40)(16,45)(18,50)(19,55)(20,52)(21,51)(22,56)(24,30)(28,34)(29,39)(31,60)(32,58)(36,57)(37,63)(38,48)(43,62)(46,64)(47,61)(49,54)(53,59)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314220
maximal normal subgroup of	#72314220

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53992);