#493246

order	512
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#508	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

712 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=ci,d^a=di,e^a=eh,h^a=hi,e^b=eh,h^b=hi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400442);