#147481

order	512
nilpotency class	4
isoclinism	isoclinic class of #13033
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(5,6,10,17,12,16,11,32)(7,18,21,46,22,45,33,42)(8,14,25,39,24,41,31,40)(9,26,28,43,30,56,29,15)(13,27,36,51,35,57,37,20)(19,44,49,63,48,66,50,67)(23,52,55,38,54,64,58,65)(34,59,60,68,61,53,62,47);(1,2,3,4)(5,7)(6,13)(8,19)(9,27)(10,21)(11,33)(12,22)(14,34)(15,42)(16,35)(17,36)(18,26)(20,29)(23,53)(24,48)(25,49)(28,51)(30,57)(31,50)(32,37)(38,63)(39,60)(40,62)(41,61)(43,46)(44,52)(45,56)(47,55)(54,59)(58,68)(64,66)(65,67);(5,8,11,25,12,24,10,31)(6,14,17,40,16,41,32,39)(7,19,33,49,22,48,21,50)(9,23,29,55,30,54,28,58)(13,34,36,62,35,61,37,60)(15,38,26,52,43,65,56,64)(18,44,46,67,45,66,42,63)(20,47,57,59,51,68,27,53)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,d^b=dh,h^b=hi,e^c=ei,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, 54677);