#147474

order	512
nilpotency class	4
isoclinism	isoclinic class of #13005
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1874
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10931
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

717 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)(3,14)(4,20,23,39)(5,25)(6,11)(7,13)(8,16)(9,22,35,24)(10,28)(12,27)(15,43,46,63)(17,41)(18,36)(19,42)(21,52,55,34)(26,31)(29,37)(30,58,59,53)(32,48)(33,49)(38,57)(40,45,62,47)(44,50,64,60)(51,54,61,56);(1,3,6,17,7,19,27,18)(2,8,12,32,13,33,11,31)(4,15,24,47,23,46,22,45)(5,26,28,49,29,48,57,16)(9,30,20,50,35,59,39,60)(10,36,25,42,38,41,37,14)(21,53,56,64,55,58,54,44)(34,43,51,62,52,63,61,40);(1,4,6,24,7,23,27,22)(2,9,11,39,13,35,12,20)(3,15,17,47,19,46,18,45)(5,21,28,56,29,55,57,54)(8,30,31,60,33,59,32,50)(10,34,37,61,38,52,25,51)(14,40,41,63,42,62,36,43)(16,44,26,53,49,64,48,58)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314266, #72314281
maximal normal subgroup of	#72314266, #72314281

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54670);