#413325

order	512
nilpotency class	3
isoclinism	isoclinic class of #398436
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1178 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,d^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi

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, 320521);