#488625

order	512
nilpotency class	4
isoclinism	isoclinic class of #488539
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#12110	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 80 points:

(4,11)(6,12)(7,14)(9,15)(17,18)(19,31)(20,37)(21,41)(22,30)(23,29)(24,27)(25,33)(26,39)(28,45)(32,62)(34,55)(35,61)(36,60)(38,56)(40,58)(42,50)(43,53)(44,54)(46,57)(47,52)(48,75)(49,68)(51,67)(59,64)(63,70)(65,80)(66,79)(69,78)(71,73)(72,77)(74,76);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19,24,36)(18,25,27,49)(20,32,40,65)(21,42,46,67)(22,35,23,34)(26,48,52,77)(28,55,54,61)(29,51,30,50)(31,57,60,41)(33,44,68,45)(37,70,58,79)(38,72,43,75)(39,69,47,64)(53,80,56,62)(59,71,78,74)(63,76,66,73);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,20)(18,26)(19,32)(21,43)(22,47)(23,39)(24,40)(25,48)(27,52)(28,56)(29,58)(30,37)(31,59)(33,66)(34,69)(35,64)(36,65)(38,46)(41,74)(42,75)(44,73)(45,76)(49,77)(50,70)(51,79)(53,54)(55,62)(57,71)(60,78)(61,80)(63,68)(67,72);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(18,27)(25,49)(26,52)(28,54)(29,30)(31,60)(37,58)(41,57)(48,77)(50,51)(53,56)(55,61)(59,78)(62,80)(70,79)(71,74)

Polycyclic presentation:

2^9|b^2=i,d^2=hi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395821);