#513079

order	512
nilpotency class	5
isoclinism	isoclinic class of #512941
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#2215	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

321 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,c^b=ci,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420275);