#146660

order	512
nilpotency class	4
isoclinism	isoclinic class of #12778
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1790
commutator	C₂×Q₈	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10745
prime radical	#3449	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

966 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317333, #72317352, #72317364
maximal normal subgroup of	#72317333, #72317352, #72317364

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53856);