#498316

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×#509	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

623 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)(5,6,10,16,12,18,34,17)(7,19,22,40,25,47,24,13)(8,26,28,42,30,61,29,63)(9,15,32,45,33,46,11,35)(14,41,43,62,44,36,48,27)(20,51,53,64,55,67,59,60)(21,39,31,50,23,57,56,38)(37,54,49,58,66,68,65,52);(1,3,4,2)(5,7,12,25)(6,13,18,40)(8,20,30,55)(9,31,33,56)(10,24,34,22)(11,23,32,21)(14,37,44,66)(15,39,46,57)(16,47,17,19)(26,60,61,64)(27,54,62,68)(28,59,29,53)(35,50,45,38)(36,58,41,52)(42,67,63,51)(43,65,48,49);(5,8)(6,14)(7,20)(9,27)(10,28)(11,36)(12,30)(13,37)(15,42)(16,43)(17,48)(18,44)(19,49)(21,52)(22,53)(23,58)(24,59)(25,55)(26,35)(29,34)(31,54)(32,41)(33,62)(38,64)(39,67)(40,66)(45,61)(46,63)(47,65)(50,60)(51,57)(56,68);(5,9,10,32,12,33,34,11)(6,15,16,45,18,46,17,35)(7,21,22,31,25,23,24,56)(8,27,28,41,30,62,29,36)(13,38,19,39,40,50,47,57)(14,42,43,61,44,63,48,26)(20,52,53,54,55,58,59,68)(37,64,49,67,66,60,65,51)

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,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405512);