#496496

order	512
nilpotency class	4
isoclinism	isoclinic class of #481393
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

788 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=ei,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^b=di,f^c=fh,g^c=ghi,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, 403692);