#512913

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂²	C₂²×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|b^2=e,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420109);