#501515

order	512
nilpotency class	4
isoclinism	isoclinic class of #501052
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

590 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)(3,7)(4,5)(6,8)(9,10,14,19,33,46,65,60,13,18,32,45,15,20,35,48)(11,21,24,51,25,41,57,71,23,16,37,39,26,40,58,70)(12,27,30,49,36,67,66,43,29,61,64,44,31,47,34,17)(22,52,54,69,28,62,63,38,53,68,72,42,55,59,56,50);(2,5)(6,7)(9,11)(10,16)(12,28)(13,23)(14,24)(15,26)(17,42)(18,21)(19,39)(20,41)(22,31)(25,33)(27,59)(29,55)(30,63)(32,37)(34,54)(35,58)(36,53)(38,44)(40,46)(43,69)(45,51)(47,68)(48,71)(49,50)(52,67)(56,64)(57,65)(60,70)(61,62)(66,72);(9,12,13,29)(10,17,18,43)(11,22,23,53)(14,34,32,66)(15,36,33,31)(16,38,21,50)(19,47,45,67)(20,49,46,44)(24,56,37,63)(25,55,26,28)(27,60,61,48)(30,65,64,35)(39,62,51,59)(40,69,41,42)(52,71,68,70)(54,57,72,58);(1,3,4,8)(2,6,5,7)(10,18)(16,21)(17,43)(19,45)(20,46)(27,61)(38,50)(39,51)(40,41)(42,69)(44,49)(47,67)(48,60)(52,68)(59,62)(70,71)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408711);