#494872

order	512
nilpotency class	4
isoclinism	isoclinic class of #33595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=i,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^c=fhi,h^c=hi

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, 402068);