#146404

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₈×C₂²⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

888 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53600);