#7604480

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563333
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#532	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=h,d^2=gi,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei

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, 7511676);