#494866

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

772 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402062);