#504261

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	#576
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

500 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=f,d^2=ei,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, 411457);