#512401

order	512
nilpotency class	4
isoclinism	isoclinic class of #511963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

739 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,g^d=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419597);