#419670

order	512
nilpotency class	3
isoclinism	isoclinic class of #417665
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1236 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326866);