#498319

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
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×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

612 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405515);