#425584

order	512
nilpotency class	3
isoclinism	isoclinic class of #417622
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
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

1238 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)(3,6)(4,8)(5,7)(9,10,16,22)(11,17,26,40)(12,18,29,43)(13,31,34,45)(14,21,15,20)(19,32,46,33)(23,37,50,64)(24,52,54,66)(25,47,35,39)(27,56,51,41)(28,58,60,70)(30,48,36,44)(38,55,65,53)(42,62,69,61)(49,67,71,68)(57,63,59,72);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,27)(13,24)(14,35)(15,25)(16,26)(18,41)(19,38)(20,47)(21,39)(22,40)(23,30)(28,59)(29,51)(31,52)(32,55)(33,53)(34,54)(36,50)(37,44)(42,68)(43,56)(45,66)(46,65)(48,64)(49,61)(57,60)(58,63)(62,71)(67,69)(70,72);(1,4,5,3)(2,6,7,8)(9,12)(10,18)(11,23)(13,28)(14,36)(15,30)(16,29)(17,37)(19,42)(20,48)(21,44)(22,43)(24,49)(25,51)(26,50)(27,35)(31,58)(32,62)(33,61)(34,60)(38,63)(39,56)(40,64)(41,47)(45,70)(46,69)(52,67)(53,57)(54,71)(55,59)(65,72)(66,68);(9,13,16,34)(10,19,22,46)(11,24,26,54)(12,28,29,60)(14,33,15,32)(17,38,40,65)(18,42,43,69)(20,31,21,45)(23,49,50,71)(25,55,35,53)(27,57,51,59)(30,62,36,61)(37,63,64,72)(39,66,47,52)(41,67,56,68)(44,70,48,58)

Polycyclic presentation:

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

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, 332780);