#493123

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	C₂⁴	#131
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

764 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:

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

Polycyclic presentation:

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

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, 400319);