#493213

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×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	C₂×#135	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

752 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:

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

Polycyclic presentation:

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

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, 400409);