#147057

order	512
nilpotency class	4
isoclinism	isoclinic class of #12826
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1804
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10850
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

525 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)(4,20,23,54)(5,25)(6,12)(7,31)(8,13)(9,16)(10,22,37,24)(11,29)(15,44,47,64)(17,38)(18,42)(19,43)(21,39,27,36)(26,34)(28,40)(30,59)(32,55,63,60)(33,50)(35,51)(41,46,61,48)(45,52,49,62)(53,56,58,57);(1,3,6,18,8,19,7,17)(2,9,12,34,13,35,31,33)(4,15,22,48,23,47,24,46)(5,26,28,51,30,50,29,16)(10,32,20,52,37,63,54,62)(11,38,25,14,40,42,59,43)(21,55,56,49,27,60,57,45)(36,64,53,61,39,44,58,41);(1,4)(2,10,13,37)(3,15)(5,27)(6,22)(7,24)(8,23)(9,32,35,63)(11,39,40,36)(12,20,31,54)(14,41,43,61)(16,49)(17,46)(18,48)(19,47)(21,30)(25,58,59,53)(26,60)(28,57)(29,56)(33,62,34,52)(38,44,42,64)(45,51)(50,55)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54253);