#495802

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3054	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

656 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)(5,6,10,16,11,15,31,41,12,17,32,42,9,29,30,57)(7,18,22,45,23,34,50,59,19,43,46,65,21,13,33,36)(8,14,26,39,27,38,55,63,28,40,56,64,25,53,54,68)(20,44,48,66,49,37,61,62,24,51,52,67,47,35,58,60);(5,7,9,21,12,19,11,23)(6,13,15,18,17,34,29,43)(8,24,25,49,28,20,27,47)(10,22,30,33,32,46,31,50)(14,37,38,51,40,35,53,44)(16,36,41,45,42,59,57,65)(26,52,54,61,56,48,55,58)(39,62,63,67,64,60,68,66);(2,4)(7,19)(13,34)(18,43)(20,24)(21,23)(22,46)(33,50)(35,37)(36,59)(44,51)(45,65)(47,49)(48,52)(58,61)(60,62)(66,67);(5,8,10,26,11,27,31,55,12,28,32,56,9,25,30,54)(6,14,16,39,15,38,41,63,17,40,42,64,29,53,57,68)(7,20,22,48,23,49,50,61,19,24,46,52,21,47,33,58)(13,35,36,60,18,44,45,66,34,37,59,62,43,51,65,67)

Polycyclic presentation:

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

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, 402998);