#508031

order	512
nilpotency class	4
isoclinism	isoclinic class of #508016
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

989 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)(11,17)(12,28)(13,19)(14,35)(15,21)(16,22)(18,31)(20,38)(23,53)(24,41)(25,57)(26,52)(27,43)(29,61)(30,62)(32,49)(33,63)(34,51)(36,67)(37,45)(39,50)(40,54)(42,59)(44,64)(46,65)(47,66)(48,56)(55,71)(58,68)(60,70)(69,72);(2,5)(6,8)(9,11)(10,17)(12,29)(13,24)(14,36)(15,26)(16,27)(18,44)(19,41)(20,48)(21,52)(22,43)(23,32)(25,39)(28,61)(30,55)(31,64)(33,60)(34,59)(35,67)(37,54)(38,56)(40,45)(42,51)(46,69)(47,68)(49,53)(50,57)(58,66)(62,71)(63,70)(65,72);(1,3)(2,6)(4,7)(5,8)(9,12,16,33)(10,18,22,47)(11,23,27,55)(13,30,15,32)(14,37,39,65)(17,40,43,69)(19,45,21,46)(20,49,51,62)(24,29,26,60)(25,58,36,64)(28,50,63,35)(31,34,66,38)(41,68,52,44)(42,70,48,61)(53,67,71,57)(54,56,72,59);(3,7)(6,8)(9,13,16,15)(10,19,22,21)(11,24,27,26)(12,30,33,32)(14,34,39,38)(17,41,43,52)(18,45,47,46)(20,35,51,50)(23,29,55,60)(25,56,36,59)(28,62,63,49)(31,37,66,65)(40,68,69,44)(42,57,48,67)(53,61,71,70)(54,58,72,64)

Polycyclic presentation:

2^9|c^2=i,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=de,f^c=fh,g^c=gi,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, 415227);