#501625

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
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	#33589	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

649 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)(5,6,11,17)(7,20,24,49)(8,26,30,14)(9,32,34,46)(10,19,35,16)(12,18,33,15)(13,25,40,23)(21,52,56,51)(22,47,58,39)(27,62,63,68)(28,61,64,45)(29,44,31,50)(36,66,57,41)(37,43,53,48)(38,55,67,59)(42,54,60,65);(5,7,10,23,11,24,35,25)(6,13,16,20,17,40,19,49)(8,27,29,54,30,63,31,65)(9,22,33,57,34,58,12,36)(14,42,44,68,26,60,50,62)(15,39,32,41,18,47,46,66)(21,53,55,64,56,37,59,28)(38,48,51,61,67,43,52,45);(2,4)(5,8)(6,14)(7,21)(9,28)(10,29)(11,30)(12,37)(13,38)(15,43)(16,44)(17,26)(18,48)(19,50)(20,51)(22,54)(23,55)(24,56)(25,59)(27,36)(31,35)(32,45)(33,53)(34,64)(39,60)(40,67)(41,62)(42,47)(46,61)(49,52)(57,63)(58,65)(66,68);(5,9,10,33,11,34,35,12)(6,15,16,32,17,18,19,46)(7,22,23,57,24,58,25,36)(8,28,29,53,30,64,31,37)(13,39,20,41,40,47,49,66)(14,43,44,45,26,48,50,61)(21,54,55,63,56,65,59,27)(38,60,51,62,67,42,52,68)

Polycyclic presentation:

2^9|a^2=f,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,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408821);