#502916

order	512
nilpotency class	4
isoclinism	isoclinic class of #501244
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	#13336	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

816 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)(11,23)(12,29)(13,19)(14,21)(15,20)(16,39)(17,25)(18,32)(22,37)(24,55)(26,43)(27,42)(28,44)(30,63)(31,61)(33,62)(34,68)(35,50)(36,49)(38,54)(40,52)(41,69)(45,70)(46,65)(47,67)(48,66)(51,56)(53,58)(57,71)(59,72)(60,64);(2,5)(6,7)(9,11)(10,17)(12,30)(13,25)(14,27)(15,38)(16,40)(18,45)(19,23)(20,43)(21,51)(22,52)(24,33)(26,36)(28,37)(29,60)(31,64)(32,59)(34,69)(35,56)(39,44)(41,47)(42,50)(46,72)(48,55)(49,54)(53,68)(57,66)(58,67)(61,63)(62,71)(65,70);(9,12,15,33,16,34,35,65,13,31,36,66,37,67,14,32)(10,18,21,47,22,48,49,61,19,46,50,68,39,62,20,29)(11,24,56,64,28,59,38,69,25,57,27,30,40,70,26,58)(17,41,54,72,44,60,51,55,23,53,43,45,52,63,42,71);(1,3,4,8)(2,6,5,7)(10,19)(17,23)(18,46)(20,49)(21,50)(22,39)(29,61)(41,53)(42,51)(43,54)(44,52)(45,72)(47,68)(48,62)(55,71)(60,63)

Polycyclic presentation:

2^9|c^2=fh,d^2=ei,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=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, 410112);