#497889

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

602 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)(7,19)(8,26)(9,29)(10,14)(11,39)(12,18)(13,22)(15,33)(16,28)(17,37)(20,43)(21,56)(23,49)(24,52)(25,53)(27,44)(30,64)(31,66)(32,48)(34,47)(35,41)(36,50)(38,45)(40,65)(42,55)(46,58)(51,57)(54,68)(59,67)(60,63)(61,62);(5,7)(6,13)(8,20)(9,30)(10,35)(11,23)(12,25)(14,19)(15,46)(16,49)(17,41)(18,43)(21,31)(22,28)(24,37)(26,52)(27,58)(29,63)(32,59)(33,54)(34,65)(36,60)(38,55)(39,42)(40,44)(45,53)(47,51)(48,56)(50,68)(57,62)(61,67)(64,66);(5,8,11,10,12,28,37,38)(6,14,17,26,18,45,39,16)(7,20,23,35,25,22,24,55)(9,27,32,36,34,31,33,62)(13,19,41,52,43,53,42,49)(15,44,47,61,48,66,29,50)(21,54,57,30,58,59,60,65)(40,51,67,56,64,63,68,46);(5,9)(6,15)(7,21)(8,27)(10,36)(11,32)(12,34)(13,40)(14,44)(16,50)(17,47)(18,48)(19,51)(20,54)(22,59)(23,57)(24,60)(25,58)(26,61)(28,31)(29,39)(30,35)(33,37)(38,62)(41,67)(42,68)(43,64)(45,66)(46,49)(52,56)(53,63)(55,65)

Polycyclic presentation:

2^9|a^2=g,c^2=f,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405085);