#505229

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
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	#25803	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

674 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)(11,26)(12,32)(13,38)(14,43)(15,45)(16,23)(17,25)(18,28)(19,37)(20,39)(21,33)(22,34)(24,44)(27,50)(29,48)(30,56)(31,47)(35,64)(36,49)(40,53)(41,51)(42,69)(46,59)(52,62)(54,63)(55,57)(58,72)(60,71)(61,65)(66,67)(68,70);(2,5)(6,8)(9,11,15,29,16,31,14,28,17,12,33,37,44,30,34,36)(10,18,22,47,24,48,21,26,25,49,43,56,23,19,45,32)(13,27,40,58,41,60,39,57,42,35,62,66,70,59,63,65)(20,46,51,67,53,64,38,55,54,71,68,72,52,50,69,61);(1,3,4,7)(2,6,5,8)(9,12,34,28,44,31,33,29,17,11,14,36,16,30,15,37)(10,19,45,56,23,49,43,26,25,48,21,47,24,18,22,32)(13,35,63,57,70,60,62,58,42,27,39,65,41,59,40,66)(20,50,69,72,52,71,68,55,54,64,38,67,53,46,51,61);(2,5)(6,8)(9,13,33,62,16,41,34,63,17,42,15,40,44,70,14,39)(10,20,43,68,24,53,45,69,25,54,22,51,23,52,21,38)(11,27,37,66,31,60,36,65,12,35,29,58,30,59,28,57)(18,46,56,72,48,64,32,61,49,71,47,67,19,50,26,55)

Polycyclic presentation:

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

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, 412425);