#148273

order	512
nilpotency class	4
isoclinism	isoclinic class of #148273
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11087
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

786 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)(3,13)(4,18,22,9)(5,23)(6,12)(7,11)(8,15)(10,26)(14,46,49,43)(16,52)(17,44)(19,56,53,62)(20,35,24,37)(21,41,28,36)(25,42)(27,45)(29,38)(30,39)(31,48,61,64)(32,51)(33,54)(34,57,55,59)(40,50)(47,60,58,63);(1,3,5,15,7,17,26,51)(2,8,10,13,11,32,23,44)(4,19,20,57,22,53,24,59)(6,27,29,50,30,16,25,54)(9,34,35,62,18,55,37,56)(12,40,42,45,39,33,38,52)(14,47,48,28,49,58,64,21)(31,60,43,41,61,63,46,36);(1,4)(2,9)(3,14)(5,24)(6,28)(7,22)(8,31)(10,37)(11,18)(12,41)(13,43)(15,48)(16,53)(17,49)(19,27)(20,26)(21,30)(23,35)(25,58)(29,47)(32,61)(33,55)(34,40)(36,39)(38,63)(42,60)(44,46)(45,62)(50,57)(51,64)(52,56)(54,59)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55469);