#505187

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

634 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,24)(12,31)(13,37)(14,22)(15,21)(16,20)(17,23)(18,27)(19,35)(25,43)(26,62)(28,47)(29,44)(30,45)(32,48)(33,49)(34,52)(36,50)(38,53)(39,55)(40,54)(41,57)(42,51)(46,64)(56,60)(58,65)(59,70)(61,71)(63,72)(66,68)(67,69);(2,5)(6,8)(9,11,15,28,17,30,14,27)(10,18,22,45,23,47,21,24)(12,32,35,63,36,67,34,58)(13,26,39,64,40,29,16,41)(19,48,31,65,52,69,50,72)(20,44,54,46,55,62,37,57)(25,42,60,38,61,68,59,33)(43,49,70,66,71,53,56,51);(1,3,4,7)(2,6,5,8)(9,12,15,35,17,36,14,34)(10,19,22,31,23,52,21,50)(11,25,28,60,30,61,27,59)(13,38,39,68,40,33,16,42)(18,43,45,70,47,71,24,56)(20,53,54,51,55,49,37,66)(26,63,64,67,29,58,41,32)(44,48,46,65,62,69,57,72);(2,5)(6,8)(9,13,14,16,17,40,15,39)(10,20,21,37,23,55,22,54)(11,26,27,41,30,29,28,64)(12,33,34,68,36,38,35,42)(18,44,24,57,47,62,45,46)(19,49,50,51,52,53,31,66)(25,58,59,67,61,63,60,32)(43,69,56,65,71,48,70,72)

Polycyclic presentation:

2^9|b^2=f,c^2=ei,d^2=fi,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412383);