#466145

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1013 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:

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

Polycyclic presentation:

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

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, 373341);