#509697

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

536 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)(9,10,14,21)(11,24,27,18)(12,30,33,70)(13,36,38,53)(15,41,40,69)(16,23,17,46)(19,37,55,26)(20,25,58,65)(22,43,61,44)(28,54,32,71)(29,52,45,51)(31,60,68,59)(34,49,35,57)(39,48,47,63)(42,62,66,56)(50,67,64,72);(2,5)(3,8)(9,11,14,27)(10,18,21,24)(12,31,33,68)(13,37,38,26)(15,28,40,32)(16,45,17,29)(19,53,55,36)(20,57,58,49)(22,50,61,64)(23,52,46,51)(25,35,65,34)(30,59,70,60)(39,66,47,42)(41,71,69,54)(43,72,44,67)(48,62,63,56);(9,12,14,33)(10,19,21,55)(11,25,27,65)(13,32,38,28)(15,42,40,66)(16,35,17,34)(18,48,24,63)(20,54,58,71)(22,49,61,57)(23,62,46,56)(26,43,37,44)(29,68,45,31)(30,69,70,41)(36,52,53,51)(39,67,47,72)(50,60,64,59);(1,3,4,8)(2,6,5,7)(9,13,14,38)(10,20,21,58)(11,26,27,37)(12,32,33,28)(15,31,40,68)(16,39,17,47)(18,49,24,57)(19,54,55,71)(22,48,61,63)(23,59,46,60)(25,43,65,44)(29,66,45,42)(30,51,70,52)(34,72,35,67)(36,41,53,69)(50,56,64,62)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=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, 416893);