#147003

order	512
nilpotency class	4
isoclinism	isoclinic class of #126701
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1735
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10842
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

505 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)(4,23,26,55)(5,29)(6,13)(7,12)(8,33)(9,15)(10,18)(11,25,39,28)(14,32)(17,45,48,63)(19,36)(20,35)(21,52)(22,41)(24,56,59,38)(27,54,58,40)(30,37)(31,42)(34,49,60,64)(43,47,61,50)(44,51)(46,53,57,62);(1,3,7,20,8,21,32,18,9,22,31,51,6,19,5,30)(2,10,14,36,13,35,12,41,15,37,29,52,33,44,42,16)(4,17,27,49,28,50,59,46,26,48,58,64,25,47,24,57)(11,34,40,45,23,53,38,61,39,60,54,63,55,62,56,43);(1,4)(2,11,15,39)(3,17)(5,24)(6,25)(7,27)(8,28)(9,26)(10,34,37,60)(12,38,42,56)(13,23,33,55)(14,40,29,54)(16,43,41,61)(18,46)(19,47)(20,49)(21,50)(22,48)(30,57)(31,58)(32,59)(35,53,44,62)(36,45,52,63)(51,64)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54199);