#501629

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

639 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)(5,6,12,18)(7,19,25,50)(8,14,30,46)(9,31,33,48)(10,17,34,16)(11,15,37,49)(13,24,41,22)(20,52,57,66)(21,58,23,39)(26,62,63,68)(27,45,32,64)(28,51,29,44)(35,53,60,40)(36,47,54,43)(38,61,67,56)(42,59,65,55);(5,7,10,22,12,25,34,24)(6,13,16,19,18,41,17,50)(8,26,28,55,30,63,29,59)(9,21,11,35,33,23,37,60)(14,42,44,62,46,65,51,68)(15,39,31,53,49,58,48,40)(20,54,56,32,57,36,61,27)(38,47,52,64,67,43,66,45);(2,4)(5,8)(6,14)(7,20)(9,32)(10,28)(11,36)(12,30)(13,38)(15,47)(16,44)(17,51)(18,46)(19,52)(21,59)(22,56)(23,55)(24,61)(25,57)(26,35)(27,33)(29,34)(31,64)(37,54)(39,42)(40,68)(41,67)(43,49)(45,48)(50,66)(53,62)(58,65)(60,63);(5,9,10,11,12,33,34,37)(6,15,16,31,18,49,17,48)(7,21,22,35,25,23,24,60)(8,27,28,54,30,32,29,36)(13,39,19,53,41,58,50,40)(14,43,44,45,46,47,51,64)(20,55,56,63,57,59,61,26)(38,65,52,68,67,42,66,62)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=e,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408825);