#512940

order	512
nilpotency class	5
isoclinism	isoclinic class of #512939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33600	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eg,e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gh,h^a=hi,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, 420136);