#147434

order	512
nilpotency class	4
isoclinism	isoclinic class of #13247
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1972
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10927
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

677 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54630);