#492973

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1080 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)(3,4)(5,6,10,16,11,15,30,39,12,17,31,40,9,28,29,54)(7,18,22,43,23,34,49,58,20,42,48,63,21,13,32,35)(8,14,25,37,26,36,52,59,27,38,53,60,24,50,51,66)(19,41,46,62,47,56,65,67,44,61,64,68,45,33,55,57);(2,4)(5,7)(6,13,17,34)(8,19)(9,21)(10,22)(11,23)(12,20)(14,33,38,56)(15,18,28,42)(16,35,40,58)(24,45)(25,46)(26,47)(27,44)(29,32)(30,49)(31,48)(36,41,50,61)(37,57,60,67)(39,43,54,63)(51,55)(52,65)(53,64)(59,62,66,68);(5,8,10,25,11,26,30,52,12,27,31,53,9,24,29,51)(6,14,16,37,15,36,39,59,17,38,40,60,28,50,54,66)(7,19,22,46,23,47,49,65,20,44,48,64,21,45,32,55)(13,33,35,57,18,41,43,62,34,56,58,67,42,61,63,68);(7,20)(13,34)(18,42)(19,44)(21,23)(22,48)(32,49)(33,56)(35,58)(41,61)(43,63)(45,47)(46,64)(55,65)(57,67)(62,68)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,f^2=h,h^2=i,b^a=be,e^a=ei,g^a=gi,d^b=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400169);